Is there any obligation or mitzvah to wash for bread if you won't touch the bread when eating?

Question

Let's say you're holding a burger in a napkin or are wearing gloves. Is there any benefit to wash your hands before hamotzi besides the fact that you are missing out on one potential beracha?

Answer 1

Shulchan Arukh OC 163:2

המאכיל לאחרים אין צריך נטילת ידים והאוכל צריך נטילת ידים אף על פי שאחר נותן לתוך פיו ואינו נוגע במאכל והוא הדין לאוכל במגרפה (פירוש כלי שיש לו שינים, פירי"ק בלע"ז) שצריך נטילת ידים.‏
One who feeds another [bread] does not need to wash his hands, and one who eats [bread] needs to wash his hands even if someone else is putting the food in his mouth and he isn't touching the food. And similarly one who eats with a rake (meaning, a vessel that has teeth, a "firk" in [the local language]) needs to wash his hands.

Eating with gloves seems no worse than eating with a fork, so washing would be required.

Answer 2

This is the method that you use if you cannot find water before making hamotzi. However as specified by the OU you must try to wash first in order to remove the tumah from your hands. Not washing when you can is more than just missing the bracha (see below) but is considered denigrating the mitzvah and causes bad results.

To find water for washing hands before eating bread:

You must travel or walk up to 18 minutes away to find water.

If you are already traveling, you must continue up to 72 minutes (in the direction in which you are going anyway) to find water.

If you still cannot find water, cover your hands with any type of separation (gloves, sheet of plastic, bag, foil, or some other object) to keep your hands from directly contacting the bread.

[Chaburas.org goes into the details in Part 1

1. The gemara in Shabbos 14b speaks of the takana of Shlomo Hamelech in the context of Rabbinical tum'ah

2.The gemara at the end of the eighth chapter of Berachot (53b) cites a verse in Vayikra 20 which states "V'hitkadishtem v'heyitem kedoshim ki kadosh ani Hashem elokeichem"

3. The most informative source for this law comes from Chullin 105a. The gemara begins by stating that mayim rishonim (i.e. the washing done before a meal) is a mitzva, while mayim acharonim (done after the meal) is an obligation (see the Chabura on mayim acharonim for more on that topic).

The main source for such statements comes from the gemara in Sotah 4b, where there are several consequences listed that will befall a person who denigrates the practice of netilat yadayim. Based on a verse in Mishlei 6, the gemara states that anyone who eats bread without washing his hands first is considered to have slept with a prostitute. Rav Zerika is then recorded as saying that anyone who denigrates the institution of netilat yadayim will be uprooted from the world. Tosafot ask why this should be the case with netilat yadayim specifically, and not with any other Rabbinic commandment? Furthermore, the gemara in Shabbat 62b states that anyone who denigrates netilat yadayim will endure poverty! They answer that this is speaking about a person who consistently disparages this law, and not simply a one-time offender.

and Part 2 which goes into more details about which foods require this

and Part 3 which gives the halacha of what if no water is available.

This section goes into detail as to what happens when all else fails. What is one to do when there is no water to be had?

There are various rulings from the Rambam, the Tur, etc and there are those that say that one is allowed to cover the hands, those who restrict this to kohanim and terumah and those who say that it is not allowed for practical reasons. See the link for details as it is too long to quote here.