Should I remove old wood glue before reinstalling a board on a table, and how?
If you want to use wood glue again, I'd remove the old glue. You'll get a stronger bond if you apply glue to clean wood. A belt sander would do a nice job. 

If you're ok using urethane glue or project adhesive, the only concern is whether the replaced board will sit above the others. Otherwise I'd have no problem bonding to the old glue if it appears solid. 

You could also run some countersunk flat-head screws of appropriate length in from underneath to assure the security of the boards.