I have started doing triathlons but don't want to invest in a Time-Trial bike until later. What are the problems with converting a road bike to a Time-Trial bike?
The geometry is the major problem with converting a road bike into a TT/Tri bike, road bikes have you sitting much farther back relative to the contact points on the handlebars.
You may be able to offset this with a different seatpost that puts you farther forward (assuming saddle is adjusted all the way towards the front).
Other changes may include a shorter stem, different saddle.
You may be able to make your road bike work, just don't expect to be comfortable mid ride switching from the aerobars to the hoods/drops. You have to pick one and stick with it.
The specific problems will depend on your current road bike. You can use clip on aero bars and a nice set of deep section wheels to make a considerable performance improvement over a normal road bike setup.
It will never be as fast as a proper tri bike, but as an intermediate step its a good compromise. The 2 main things you will miss are aerodynamics and shifters on the aero bars.
When you do come to upgrade, as noted by @Frisbee, TT and Tri bikes are not the same thing, Tri bikes can offer better aerodynamics since they don't need to conform, to UCI regulations.