Can airliners operate without an operating APU?
Yes, you can operate without an APU.  If the APU is inoperative you will need to write it up and coordinate with maintenance and dispatch to operate under the stipulations of the MEL.  This may restrict certain kind of operations but if we assume a route completely over land this has no effect other than needed ground services.

The requirements for no-APU is simply a ground start "huffer" cart at your departure airport.  Ideally you want one at the destination as well but if not available you can possibly coordinate leaving an engine running during the turn.  More generally though, if the destination does not have access to a huffer cart you will get an aircraft swap and the non-APU aircraft will be assigned to a route that does have those ground services available.

The general process of a non-APU flight is keeping ground power and ground air conditioning hooked up to the airplane for pre-flight activity and boarding.  Once the airplane is closed up the ground crew will coordinate with you to provide air via the ground start cart and you'll start one or both of your engines.  After you have an engine running, you'll disconnect all of the ground equipment and push back.  During the taxi, if you only started one engine earlier you'll perform a crossbleed start to start the other engine.  The takeoff engine bleed and pack configuration may be different for a non-APU takeoff but once you are airborne and set the bleeds and packs for post-takeoff you will be operating exactly the same way you would normally operate and the differences cease.

On arrival the only differences are to make sure you keep an engine running at the gate until ground power is hooked up.  

The biggest downside to operating without an APU is the lack of decent air conditioning during taxi and ground operations.  



Other considerations for operating without an APU are airplane specific.  For example, the EMB-145 has two generators per engine and one on the APU for a total of five generators.  The airplane will revert to "essential" power if less than 3 generators are available which means you will lose certain displays and systems that become unpowered.  This means if you had an engine failure during flight without an APU you will be in essential power and have to land in that condition because you cannot start the APU to get back up to 3 generators operating.