The reason I think Tyr's Sword is better is that it increases the area where two-handers are missing out (AC, vs shield-users, basically giving them the effect of a shield without having to give up the damage) while using up a resource Fighters generally don't use at all (Concentration), making your character more well-rounded. Add to that that since Concentration is kept based on two things: Whether or not you get hit, which the AC bonus makes less likely to happen; and if you get hit the Concentration checks based on Con, which a Fighter will generally have a lot of. Oh, and it uses a bonus actions which Fighters do not generally make heavy use of, and is available on a by-battle basis, so there's really no reason not to slap it on at the start of every combat unless you need the bonus action to jump into reach of the enemy or something.
Everburn on the other hand just adds 1d4 fire. The sword is essentially no different from any dipped greatsword, except for the part where it's unquenchingly dipped.
Sorry for lingering on this, I just wanted to justify myself
