Finned Baseboard and Slab Leak
 

I've got a leak in the concrete slab where the copper tube is routed under a sliding door for my hydronic baseboard.

Can I route the tube over the door and though the drop ceiling, or would it be better to cut the concrete and replace the copper?

Welcome TW:
If the installer didn't know to keep the copper tube seperate from the concrete it would be best to run it overhead. Concrete and copper just aren't compatible and the copper looses every time.
When you run the line into the drop ceiling, make one end higher by 1/4" and put in an automatic air vent in. That will now be the highest point of the system and it needs an air bleed to get the lines to run full and heat well.
Glenn

Hmmmm
 

Glenn always knows just what to say.:)
And another point is ...if you have any more lines in the concrete, you might as well go ahead and change those out. The system will need to be drained anyway so now is the time.
Good luck.

Thank you, thank you, now I have a plan. Follow-up Question: Is there any way of neatly "trimming in" the copper that will run vertically up each side of the sliding door?

Yes, you could use a 2 x 4 finish grade and rout out a channel in it for the pipe. You will probably be using 3/4" copper and the outside measurement of even the fittings is 1" while the tube is 7/8" od.
Some trim around the top and bottom could make it look like load-bearing posts.
Glenn