Can I use TBE to elute a plasmid shipped on filter paper? I'm receiving a plasmid shipped to me dried on sterile filter paper, as described in this question. This protocol and many others like it call for eluting the DNA from the paper using TE buffer. My question is can I also use TBE to elute, as I already have a bunch on-hand? And in general, what concentration should the buffer be as I never see that mentioned in any of the protocols (including the AddGene version).

In my experience it does not matter if you use water, TBE or TE buffer. Even with a little EDTA in it, it will not mess up enzymatic reactions as the concentration is way too low. TBE/TE buffer usually contains 1mM of EDTA which can chelate 1mM of bivalent cations. Since you not use the DNA undiluted (which also binds a lot of magnesium needed for most enzymes btw.) this really doesn't affect you in reality (been there, done that multiple times).

Additionally, I would recommend not using this DNA directly but transform it into bacteria first and then do a proper miniprep so you have enough material to work with (maybe even do a few restriction cuts and run it on a gel, to ensure the right identity) and a stock you can go back to if something goes wrong.