Would tactics used by Roman legions be relevant against tactics used in the period from 1300-1400? If you had the time and money to train and equip an army to use Roman tactics (things like men dropping back through the ranks to rest and allowing you to cycle fresh troops forward and throwing pilum before switching to a modern gladius equivalent) and used developments in equipment and technology, would it be able to outmatch other feudal forces of the same size? Are there any ways to improve it?

Probably not. See, the thing about Roman infantry tactics is that they were excellent against infantry, whether we’re talking the classic Hellenic phalanx or the less well-organized forces of the Gauls or the Goths, because they combined flexibility (through the cohort system) and discipline. 

But they were never any good against cavalry, whether it was Hannibal’s Numidians or the Parthian horse archer and cataphract combo. Thus, by the time that we get to the 3rd century and the Romans are facing new military threats who are almost entirely cavalry, the Romans themselves stopped using Roman infantry tactics. The legions might have been peerless infantry in their time, but as foot soldiers, they couldn’t get to the threatened parts of the empire in time. And on the battlefield, they were extremely vulnerable to being flanked and overrun by cavalry. 

And that would have been the problem if a Roman legion had suddenly appeared in 14th century Europe. Take it step by step: the legions would start by throwing pila, the Europeans would answer with massed archers firing into the close ranks of the legion. Then the Romans advance with gladii up against dismounted men-at-arms and get stopped in their tracks, unable to penetrate their full plate armor. Then the auxiliary cavalry go up against armored knights who’ve learned to couch lances and use stirrups, and get butchered like sheep. Then the knights circle back around and hit the legion in the rear, and it’s all over.