Can I use TBE to elute a plasmid shipped on filter paper?
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.