The sentence is 水道の水を飲んでも大丈夫です. The English translation is, "You can drink the water from the tap." I have learned ～てもいいです is for giving or asking for permission. Is this just the same thing but possibly more casually? Switching out the いい for 大丈夫? If not, I'm not sure what else a te form verb + も大丈夫 could mean.
I think that the translation is good, but it fails to capture what the grammar is doing here. Here is what (in my opinion) is a more literal translation, though slightly less natural to English speakers.
Even if you drink water from the tap, its okay.
The base て + も grammar is often translated to
even; even if; even though. You can study more about this grammar here. Suffice it to say that there are lot of potential uses beyond what you are already using it for.
To keep the answer from getting too long, I'll only talk about situations where permission is given. Just know that the use for this grammar is much more expansive, and I recommend that you follow the link.
I have learned ～てもいいですfor giving or asking for permission. Is this just the same think but possibly more casually? Switching out the いい for 大丈夫?
You are right to translate the two to have the same general meaning. However, I can't really say whether or not it is more casual. It's a different word, so there's just a slightly different meaning. To my understanding ～てもいい means that (verbing) is acceptable. ～ても大丈夫 carries more of a feeling that it is okay to (verb). In translation, these subtleties are often lost so the end result often ends up being the same.
Here are some ways that you can give permission using this grammar (apologies for the repeats):
～てもいい -- (Verbing) is acceptable.
～ても大丈夫 -- It is okay to (verb).
～ても構｛かま｝いません -- I don't care if you (verb).
～てもけっこうです -- I am okay with you (verbing). (This is more casual than the others.)