Does a player always has to have uncompleted destination tickets on-hand?
No.

If you do complete all your routes, it is often a good idea to collect more since each route completed adds to your score, however this is very dependent on the current game situation (how many routes are free, how close it is to game end, how flexible your current rail system is) as you also risk drawing new cards that you can't complete and which would penalize you. However, there is nothing mandating that you always have tickets to complete: Since the contents of your tickets is not public information nobody should know that you've completed all your tickets until game end, so such a rule would be mostly unenforceable anyway.

According to the rules, the game turn consists of taking one (and only one) of the following four actions:


Draw Train Car Cards
Claim a Route
Draw Destination Tickets
Build a Train Station


However, there is absolutely nothing in the rules which tells you to take one of these actions over the others in any situation: It is entirely your choice as to which of the four actions to take.

Literally the only case I can conceive of where you'd be forced to collect more destination tickets (aside from the ones drawn during game setup) would be if the other three actions were, for whatever reason, unavailable to you. This would basically mean that there are (a) no train cards to draw, (b) you can't afford to claim any routes, and (c) you've already deployed your station. And if this does happen to you, you've probably already lost.