I just need to take a screenshot of an area with street names visible, but without business names. Is there some option of the website to just hide them temporarily?
No, not on openstreetmap.org. But OSM is just a database, and many MANY maps are built upon that. Some of those maps definitely have that option.
I haven’t tried it personally, but try Cruiser (a Java app) with an offline map, and experiment with themes. (I use Locus Map on Android, themed offline maps as well, and I know that does what you want.)
If it’s a small screenshot you want, simply scrub out the unwanted names with a paint program. 
And another useful one for such a case (with many different renderings as well as the option of adding your own stuff) is this one: