Never resign until you have an unconditional job offer.
However, assuming this is the US, in almost every case your HR reference will be a formality. They will do no more than confirm that you worked in the position you said for the dates you said. Nor will they do less, by which I mean it is extremely unlikely that they will tell your new employers anything bad about you. So yes, you should give them permission to contact your HR before you resign. Don't worry, this is the normal way of doing things.
If you have particularly good relationship with your boss, you might like to tell him before the new company contacts HR. Make sure you do it informally and make it clear that you are not resigning. But this is only to be nice to him - it isn't necessary, and a good boss won't expect it. And be aware that it's a high risk thing to do. Some bosses will look to get rid of you if they think you are looking for another job.