What are the kashruth issues involved in the taking of gel-capsule medicine which does not have kosher certification?
According to Rabbi Dovid Heber, Kashrus administrator of Star-K, pills that are swallowed whole without chewing, poor-tasting liquids, and unflavored tablets may be taken, even if they are not kosher, by anybody whose life is in danger from the illness, by anybody who is so sick that they are completely unable to function properly, or by a child under six years old who is feeling discomfort.
Somebody who is just feeling discomfort, but is able to function, should consult a Rav; however, even someone who is just feeling discomfort may take a medicine where all of the ingredients contain no food, nutrition, or hydration value.
In addition, any unkosher ingredient which is batel bashishim (nullified because it makes up less than 1/60 of the total product) is acceptable in medication, even for someone who is just experiencing discomfort, and even when there are additional stringencies with regard to bitul bashishim with food.
I don't know this for a fact, so don't act on it without finding out, but I would imagine that in nearly any medication which contains unkosher ingredients, those ingredients would be bitul bashishim.
Edit, Rabbi Heber says this about gelatin capsules:
Certainly, if there is a choice of non-gelatin capsules, they are preferable, though most poskim would allow one to take them:
There is quite the machlokes (argument) as to gelatin capsules:
The Israeli Rabbanut allow it for pills as not considered food - שלא כדרך אכילה.
Other Poskim hold such capsules to be edible (כדרך אכילה) and thus should be avoided (See Tzitz Eliezer intro to vol 4).
R' Yehoshua Neuwirth (Shemiras Shabbos Kehilchasa 40 note 169 ) and R' Ovadia Yosef (Yabia Omer 12:11) are lenient.
R' Eliezer Waldenberg (Tzitz Eliezer 6:16; 7:32:8) allowed it for a non-seriouslsy ill patient (חולה שאין בו סכנה)