Do skills or items that increase fire rate also affect melee attack speed? (e.g. Krieg the Psycho's Buzz Axe Rampage)

When throwing the Buzz Axe while Buzz Axe Rampaging, yes, Fire Rate boosts (from items and skills) affect how fast you can throw Buzz Axes.

Discussions on Buzz Axe throwing fire rate boosts from the Gearbox Software forums:

• Fire Rate increase help with Axe Throw?
• Just in case anyone didn't know, fire boosts apply to thrown buzzaxes.
• Is there a Throw-Axe focused build for Krieg? Viable?
• Apparently fire rate boosts afect how fast you can throw the buzzaxe

The thrown axe counts as a pistol, so pistol fire rate is what you are looking for. I run with the Sheriff's Badge exclusively now and farmed the 56% one just for this reason. With no Blood Trance I get 16-17 throws up from the normal 11-12. Unfortunately, the pistol damage boost on the Badge doesn't effect the thrown axe :/

While writing this it crossed my mind to test Embrace the Pain, and that does NOT boost the thrown axe. This is an exception, all other general or pistol fire rate increases should work. Elemental Elation DOES work.

I second the answers from others, the primary melee doesn't change in speed.