Is it possible to remove turn limits in multi-player games?

You won't be able to change it now that the game is started. The option is set at the start of a game, in the advanced options, there's a Max turns settings.

Setting it at 0 will remove the turns limit of a game. Source from Steam

Linked page 3 as it is the page with the most important information on how to see that 0 is the proper value and this is from page 4 :

1) First game. Set up game, advanced options. Max Turns option was NOT checked (this appears to be the games default configuration unless you change it). Disabled all victory conditions. Started game, checked game details from menu, it listed "Time" as a victory condition (peculiar, but proceeded nonetheless). On Turn 400 the game abruptly ended (was playing vs 1 AI on easy, i was just ending turns over and over not doing anything... and again, all victory options were turned off). Game abruptly ends with the "Time" ending (shows the box art screen, a brief voice over about time running out, game ends)

2) 2nd game. Set up game, advanced options. Max Turns checked on, set to 0 turns. All victory coniditions disabled, otherwise identical settings as last game. After 560 turns the game was / is still continuing with no end in sight, game doesn't appear to end at turn 400 as the previous game did.. thus this worked.

I made a simple web app that will let you change the game parameters of a save file, including max turns: Civilization V save editor

Just open your save file in the app and uncheck Time victory (it will automatically set Max turns to 0). Then download the save file with the changes and open it in the game.