If one is in a dark place where he can’t read the siddur without artificial light and there are no other lights available, would he be permitted to hold a smartphone, while occasionally pressing it so that the screen doesn’t go dark, in order to shed light on the siddur that he is davening Shemoneh esrei from, or would this be forbidden on account of holding something other than a siddur during Shemoneh esrei
I think it's pretty clearly מותר. The הלכה is (ש"ע א"ח צו:א) that one should not hold objects during שמנה עשרי because he will be worried that he might drop them and it therefore disturbs his concentration. However, holding a סידור is permitted (צו:ב) because it's a need of the תפילה. Holding a smartphone for light should be no different; since it's a need of the תפילה one may hold it during שמנה עשרי.
However, I would like to point out from a practical standpoint that this is not such a good case. Most smartphones have flashlights, eliminating the need to tap the screen (or can be set so the screen never times out), and many smartphones also have סידורים (although I'm not certain if that would be preferred). What I wrote applies in a case it's actually necessary for the תפילה.
שולחן ערוך אורח חיים צו
כשהוא מתפלל לא יאחוז בידו תפילין ולא ספר מכתבי הקודש ולא קערה מלאה ולא סכין ומעות וככר מפני שלבו עליהם שלא יפלו ויטרד ותתבטל כוונתו...
משנה ברורה שם
ודוקא הני שאם יפלו יש בהם הפסד או יזיקו לו אבל שארי דברים מותר לאחוז וי"א דהני לאו דוקא...
Shulchan Aruch O"H 96
When praying, one should not hold Teffilin, a holy book, a full bowl, knife, coins or loaf because one is watching over it, so they will not fall, so he is distracted, and his focus is disturbed.
Mishna Brura, There
Specifically those, because if they will fall, they will be damaged or cause damage, but other object allowed, and others say, all objects are forbidden
So, for one opinion, it's obviously allowed, because the phone helps the prayer, not just not disturbing.(@Rafael's answer) IF IT'S ON FLIGHT MODE!!!, Otherwise, it's the biggest distraction exists.
About the other opinion (T"Z), it's source is the rule of not holding anything in front of you while praying:
שולחן ערוך אורח חיים צז
הנושא משאוי על כתפיו והגיע זמן תפלה פחות מד' קבין מפשילו לאחוריו ומתפלל ד' קבין מניחם על גבי קרקע ומתפלל:
Shulchan Aruch, end of O"H 97
One who carries burden on his shoulders, and it's time to pray, if it's less then four kabin (small measure of volume), he passes it behind (on his back), if it's more, he puts it on the ground and pray.
The connotation of Shulchan Aruch paragraph 97 is "things you don't do while praying, since it's not honorable".
Is holding smartphone in front you is dishonorable? This question is valid even if we accept the first opinion on paragraph 96!
Some Rabbis consider it the tool of the devil. Some Rabbis might not like a tool made for conversions in praying, making it very dishonorable ,especially in shul. Some Rabbis might allow this.
Consult your Rabbi!!
1) The T"Z explains that any object held in front is problematic, not just burden. That can be argued.
2) Aiding the prayer might nullify the dishonor of holding an object while praying (makes perfect sense), but maybe not the dishonor of holding a smartphone during prayer (depends on the Rabbi, I guess).