This would seem to be rooted in a machlokes Tannaim, between R’ Elazar son of R’ Shimon and R’ Yehoshua ben Karchah.
As recorded in Bava Metzi’a 83b, R’ Elazar was appointed as a police officer for the Romans (read: if anyone was a thief, he would arrest them and lead them to jail, where they would shortly be killed). R’ Yehoshua ben Karchah sent him a message calling him “vinegar son of wine,” i.e. wicked son of righteous, and rebuked him for informing on G-d’s nation for them to be lead to their deaths.
R’ Elazar responded that he was picking the thorns from the vineyard, i.e. he was only sending off those who truly deserved it. R’ Yehoshua responded that the Owner of the vineyard can take care of His vineyard just fine.
One day, however, R’ Elazar had a man who insulted him be arrested, as only a wicked person would be insolent enough to act like that. Nevertheless, he regretted his decision later, but it was too late. His students assured him that it was okay; after all, this man was known to regularly spend his Yom Kippurs having intercourse with a Na’arah Me’orasah (two sins in one). R’ Elazar was calmed, and he said, “If my uncertainties are this accurate, certainly my certainties.” The story goes on to show that, just to be certain, he had a liposuction performed on himself and left the fats out in the sun in the heat of the summer, with the veins still in the fats (apparently requisite for them to dissolve?), yet the fats would not be dissolved.
Rambam and Shulchan Aruch
What we see from this story is that according to R’ Elazar, one is able to turn over criminals to the authorities, as HaShem doesn’t allow a righteous person to be killed; in contrast, R’ Yehoshua held that one should not take this upon oneself, but rather, he should let HaShem clear out His own vineyard.
Rambam (Chovel U’Mazik 8:9) paskens like R’ Yehoshua to the extreme:
אָסוּר לִמְסֹר הָאָדָם בְּיַד עַכּוּ''ם בֵּין בְּגוּפוֹ בֵּין בְּמָמוֹנוֹ. וַאֲפִלּוּ הָיָה רָשָׁע וּבַעַל עֲבֵרוֹת וַאֲפִלּוּ הָיָה מֵצֵר לוֹ וּמְצַעֲרוֹ. וְכָל הַמּוֹסְרוֹ בְּיַד עַכּוּ''ם בֵּין בְּגוּפוֹ בֵּין בְּמָמוֹנוֹ אֵין לוֹ חֵלֶק לָעוֹלָם הַבָּא:
It is forbidden for a person to hand over to the idolaters, whether his life or his money, even if he’s a wicked person who does many sins, and even if he is annoying and paining him. Anyone who hands over to idolaters, whether his life or his money, has no share in the World to Come.
The Shulchan Aruch (CM 388:9) rules similarly. The Rema adds a note about certain exceptions:
אסור למסור לישראל ביד עכו"ם בין בגופו בין בממונו ואפי' היה רשע ובעל עבירות ואפי' היה מצר לו ומצערו: הגה ודוקא בדברים בעלמא אבל אם מסרו מותר למוסרו דהרי יוכל להרגו בדין במקום שיש חשש שיחזור וימסרנו (הרא"ש כלל י"ו סי' א' וב' ותשו' רשב"א סי' קפ"א) או אם אי אפשר להציל עצמו בדרך אחר אבל אם אפשר להציל עצמו בדרך אחר הוי כשנים שמסרו זה את זה וכל מי שהפסיד חבירו יותר חייב לשלם המותר בנזק שלם (מרדכי פ' הנ"ל ותשו' מיימוני הנ"ל) וכל המוסר ישראל ביד עכו"ם בין בגופו בין בממונו אין לו חלק לעולם הבא:
It is forbidden for a person to hand over to an idolater, whether his life or his money, even if he is a wicked person who does many sins, and even if he is annoying and paining him. Rema: This is only in regard to Earthly matters, but if he had previously handed someone over to the authorities, one may hand him over, since according to the law one may kill him if there is room to worry that he will continue handing over. He may also hand him over if that’s the only way to escape; however, if he is able to escape in some other fashion, then it’s like two handing each other over, in which case whoever damages more pays the difference. Whoever hands a Jew over to the idolaters, whether his life or his money, has no share in the World to Come.
So, to address your questions:
- Is there a Halachic obligation (besides for dinah d'malchuso) to report a crime to the non-Jewish (local/state/federal) authorities should one witness one? Forget dinah d’malchusa. Don’t report the crime.
- Does it matter if the crime is something illegal according to Torah? No. Still forbidden.
- Does it matter if the criminal is Jewish or not? Yes. All of the above sources discuss handing over Jews; there is no prohibition against handing over non-Jews; to them, we’re no different than any other law-abiding citizen, who would do the same.
EDIT: Thank you @DoubleAA for bringing this to my attention.
Does this apply to America?
Apparently there is a machlokes in the poskim as to whether this discussion remains in force today.
In order to discuss this in respect to today’s society, though, we must first note a few things about today’s American government:
- In general, the American government punishes exactly according to its legal code; there are no punishments given to one found innocent by the law, nor are there extra or different punishments given to one found guilty, beyond that which is stated in the relevant law codes.
- By the first amendment, punishments are given solely because one broke the law and not because of anti-Semitic motivations.
- In terms of American law, a Jew who refuses to go to Beis Din cannot be forced to go.
- Because of American law, Beis Din cannot enforce physical punishments, incarceration, or the like, nor can they exercise force in emergency situations.
- According to American law, there is no obligation to report a crime when one is not involved in it somehow. However, if one is summoned to testify, one must comply.
It should be emphasized at this point that, as per the above Rema, one must hand over someone who is a danger to society according to all opinions.
- The Aruch HaShulchan (CM 388:7) writes that in a government that gives right to property and where there is no fear of torture or death (unlike the Roman government discussed in the Gemara, which would seize property just because, and which would torture those that didn’t comply) one
should may (thanks for the catch, @DovF) report crimes. R’ Elyashiv (the above article quotes Devar Sinai 45-46) writes similarly, but notes that Chillul HaShem could be a factor which must be evaluated on a case-by-case basis.
- R’ Ezra Batzri in Dinei Mamonus 4:2:5n1 quotes the above Aruch HaShulchan and argues vehemently, that such opinions are set forth by naïve people. He holds that there are no just governments and no just prisons. So many problems can occur throughout the system that one must assume that injustices will occur, and so the rule that one may not inform still applies today. Pischei Choshen 5:12:5 in the notes there adds that prison today is such an awful place to be (studies show that 14% of inmates are assaulted in prison) that it would be forbidden to submit one to such treatment - and thus informing is forbidden.
- The issue at hand, if it’s one of fear of torture, is ultimately a question of being a rodef. According to Beis Yitzchak (YD 49:12), therefore, we can draw a distinction between cases of submitting one to a death penalty and submitting one to a monetary penalty. In the former case, it’s entirely forbidden; in the latter case, while it may be forbidden, he merely has to reimburse the fines.
- Rav Wozner (Shevet HaLevi YD 58) holds that one may inform the government when Jewish law accepts that part of secular law. For instance, one may be a tax collector, since Jews accept that they should pay taxes. In the above case in Bava Metzi’a, though, the issue is that handing them over is a life or death issue, and thus should be prohibited. It would come out, then, that according to this opinion, dinah d’malchusa is very relevant.
- R’ Moshe Feinstein, in Igros Moshe CM 1:8 and 5:9:11, writes that one may not inform, since he will be judged inconsistently with Jewish law. He notes further in CM 1:92 that one can work for the IRS since (presumably) the cheating will be detected in any event, and since there is no loss, there is no prohibition. Notice that in neither of these teshuvos does he discuss the justice of American law, clearly implying that the laws of informing still hold under today’s government.
Since everyone agrees that a danger to society may be turned over, seemingly the sources in @mbloch’s excellent answer can hold like any of these opinions.
So, to readdress the questions in the OP:
Is there a Halachic obligation (besides for dinah d'malchuso) to report a crime to the non-Jewish (local/state/federal) authorities should one witness one? Depends on your posek: according to opinion 1, yes; according to opinions 2 and 5, no; according to opinions 3 and 4, it depends.
Does it matter if the crime is something illegal according to Torah? According to opinion 1, no; it’s always permitted. According to opinions 2 and 5, no; it’s always prohibited. According to opinion 3, no; the distinction is whether it’s life-or-death or merely monetary; according to opinion 4, this depends on whether you consider dinah d’malchusa to be “illlegal according to Torah.”
Does it matter if the criminal is Jewish or not? Yes. The answer to this question remains unchanged.