Doing this can definitely Backfire!
I have seen it happen 2 times to other people and once in my own case. Basically, at first telling a school I had another offer to interview seemed to help to negotiate a little. While they were inflexible with their deadline and the next interview would come later; I asked if they could do anything about the salary and/or start up funds. They raised the salary a grand and I was happy enough with this point, but they ignored my question about whether or not anything more was available for start up funds.
When I wrote back to thank them for salary and ask if they had any news regarding the question for funds, They decided to rescend there offer!
Just the fact that I asked (and this was I thought a modest request) made the school lose interest in me as candidate; they said they no longer thought I would be happy and stay there.
The other school turned out to be horrible; they wanted me to teach 3 classes a semester between 2 satellite campuses 90 minutes apart, and to teach and run their undergraduate research lab for seniors and juniors 15-20 hours a week to help beef up their research program; it was definitely set up as a burn-out job, and paid one grand more than the original offer.
So, I guess the moral of the story is don't forget who REALLY has the leverage; academic jobs are few and far between while PhDs with good research and teaching skills are a dime a dozen. Is it any wonder that university and colleges largely feel they can take or leave us on a whim? There will always be another good candidate that easily could take our place, and who is more desperate (willing to work for less and do more work).
Is scratching and scraping to make a very modest living as an academic worth it? Is working 60-90 hours weeks so that you can make enough to meet the national average cost of living for a household really worth it?
There has got to be a better way!
So, on at least 3 occasions that I know of in recent years, schools felt they had superior leverage