Don't forget that a wider tire is also higher. Therefore you need to
check the length or space between the bigger tire and the chainstay
bridge if there is one and also the space between the tire and the
brake bridge.
I've had 1980's era steel frames, a Fiori Modena being one of them,
where the height of a 32 mm tire would cause it to almost rub the
chainstay and brake bridges. On some old steel frames the tire would
rub the chainstay bridge.
That being said, it is relatively easy to cold-set (spread the
chainstays) and Sheldon Brown's site has an article on how to do it.
Any *COMPETENT* bike shop should be able to do it for you.
Will you have enough height between the bigger tire and the brake
bridge to put a fender so that it doesn't rub the top of the tire?
Cheers