How long the application can run with desktop as runtime platform?
What all problems will be caused on using desktop as runtime platform instead of Experience portal.

It will probably run for a bit but you will not get any support if your trace log says platform desktop and you are running on the platform. The license is free. You will consume a lot of socket ports other stuff will not work as expected like CAV, conversations, CTI, and etc.

So, the application will run fine except the CAV, CTI and AEP reports.

What is the reason you want to stay with desktop mode? Are you trying to get around the licensing? Can't promise that the system won't behave erratically over time in desktop mode and we won't be able to troubleshoot or provide support unless it's running with a proper configuration. Most PDC's will not work as well as any type of communication you need to do to the platform: CAV's, platform web services, reporting. In desktop mode, we open up a socket connection to the simulator which is going to consume more ports as well as TCP/IP resources on your system. Simulation is meant for one call at a time and not for higher volume, so the behavior could change as your traffic increases. Also no guarantee that anything working today won't break in this mode in future releases.

We want to stick to Experience portal license. but application with desktop runtime is deployed. We will get chance to change only after a week. Just want to know will the application hold for a week.

The is not recommended. Remember the OD license is free.