Why would Captain Beatty want to kill himself?why would he provoke montage when he had a flamethrower?

In Fahrenheit 451, suicide is
common.  This is revealed by the nonchalance with which the technicians treat Millie
after her overdose.  It is not surprising that Beatty would just as soon die as remain
alive.

Today, we call what Beatty does "Suicide by cop." 
Beatty commits suicide by firemen.

To know why Beatty
commits suicide all you have to do is look at the society in which he lives.  Beatty is
intelligent--he's too intelligent not to know that his existence is
lacking. 

Ironically, Beatty allows himself to be killed
for the same reason Montag begins reading books--his existence sucks. 
 

The world they live in is mindless, superficial,
unemotional, unenlightening, unfulfilling. 

The real
question is:  why would anyone want to live?