Not really. While Hulk has been in Spider-Man stories a lot, there is one hero who has teamed up with Spider-Man the most outside of Johnny Storm - and that's ol horn head who has the more gritty vigilante angle going for him. Granted the Hulk is not that far behind those two.
For those interested Charlie Cox has said that he's more than interested in joining up with Spider-Man. Plus, interestingly, Holland and Bernthal are friends - they auditioned for Spider-Man and the Punisher off of each other so that famous triad (fingers crossed) might happen at some point.
As someone said, I'd liken Banner more to Doc Brown or (to bounce off of DC) almost Harrison Wells (season 2, Flash) due the fact that Peter never really knows when Banner will, and often does, Hulk out - with Mister Fantastic being the more stable father figure scientist of the two. Both basically exactly similar, just one brings more safety and less risk of a Hulk-out.
--------------------
For those wondering, at a fast glance, Spidey and Iron Man team ups (solely about them teaming up) are:
Invincible Iron Man 7 (2009)
Marvel Team Up v1: 9, 29, 48&49, 51, 72, 110
The Invincible Iron Man #234
The Amazing Spider-Man 25th Anniversary Annual
I would say my favorite line between those two is from Civil War (the comic arc), I don't know the
exact phrasing of it - but it was about Tony telling Peter he could easily see he wears his need for a father figure on his sleeve. I'd say that's a role that Tony, Reed, Bruce, and Steve all basically function as in Peter's life especially the first three.