Can the processor just turn off the interrupt request line?
It depends on the processor arhitecture, but yes, most processors can turn off interrupts.  Shutting them completely off is usually only done in special situations like booting or critical error handling.

More commonly, interrupts are blocked until the CPU returns from the interrupt handling routines.  These routines should typically do the minimum amount of processing required, then return from interrupt to finish any work (forking down to a lower level for most of the work).

An older system I'm most familiar with, the VAX architecture, allowed for multiple interrupt levels (up to 31) and when an interrupt occurred the CPU would run at an IPL level corresponding to that interrupt.  All other interrupts at or below that IPL would be blocked until the IPL was lowered below the level of pending interrupts, then the pending interrupt would be delivered.  There was also a SETIPL instruction that allowed critical system code to block certain types of interrupts while critical code executed.  On single processor systems this was an effective synchronization technique that prevented multiple streams from changing data structures in an uncontrolled manor (multiprocessor systems required using other techniques like spinlocks).  Some problem with blocking interrupts are that multiple pending interrupts would be delivered as a single event and devices with unserviced interrupts sometimes get timeout errors.  So running at elevated levels for extended periods is discouraged.