Man that sucks. If it was me I wouldnt do anything that would keep me from lifting longer than I had to. Take some time off and let it heal man. Some people would say 4 weeks maybe 6. Depending on how severe. But, Im no Dr. All I can say is what I would do.
Maybe you just strained something and all you need is just a few weeks off. I just got back in from a strain in my left rotator cuff. I had a really sharp pain when would work it. I gave it 2 weeks and good as new.
Give it a rest and then go back into it light and see how it feels. If it is hurting dont do it. Just dont do anythint to make it worse. Its not worth it.
Depends on your age, whether you have a tear, if so full thickness or partial, and thus likely impingement, or ? a/c joint, ? +/- biceps tendonitis/SLAP tear.
If you are over 35, I would see a doc, get MRI, and if no full thickness tear, would then get one visit with a physical therapist who is a weightlifter.
One visit they can show you: 1) how to do rotator cuff/scapula exercises and stretching to help prevent further injury/impingement. Be careful of some of the "external rotator exercises" you find on internet (ie cubans etc, dont do)
2) Which exercises to modify/avoid if you have impingement or RC tear (almost always supraspinatus)... (ie, no dips, upright rows, lateral flies and only do bench with elbows in and elbows should not go behind plane of back on way down).
Personally, I think everyone over 35 should be mindful of their cuff. I wish I had first time I injured my shoulder. (as an aside 30% of non-weightlifting 40 year olds have asymptomatic cuff tears, ie pretty common)
I wouldnt do anything until pain at rest is gone, and ice 20 mins 2-3 times per day will help with that.
Or you can ignore it and keep lifting like I did, and essentially try and saw your supraspinatus in half, by doing some exercises you shouldn't, failing to do some you should, and doing others incorrectly.
After finally doing things right way (for an over 35 yo with rc problems), it took about 4-6 weeks to get me back to lifting decently (though will have some mild discomfort I am told for few months).