Cross contamination could be an issue. You WANT lactobacillus for cheese making, and you don't want it for beer.
Today I'm making a batch of chevre. I did all of my wine stuff first, and put them away and under airlock. (I racked 16 gallons of wine!). Then, I started the cheese. It has its own dedicated thermometer, cheesecloth, muslin, strainers, etc.
For just straining some yogurt, you just need to have a separate grainbag type thing I'm sure. I'd be concerned that if it's grocery store cheesecloth, the holes would be too far apart anyway. I use cheesemaking cloth- it's reusable (washable) and not full of those holes that "cheesecloth" has. It's a tighter weave.