I put an old packet of yeast into my beer and it is barely fermenting. If I add a second type of yeast will the first one have any effect?
I think you should be good to add the other packet as long as you do it quickly, for several reasons, mainly, the if wort was aerated, the new yeast can still feed on this oxygen for reproduction and cell-wall building; also, the wort will start oxidizing right away which will diminish your beer's flavor if it's not consumed quickly.
Also, if yeast doesn't start reproducing and take up all the nutrients your wort is at a very high risk of getting infected with bacteria, and then you won't have any beer at all.
I hope you can get that beer going soon! :)
First, how to you know that the first yeast is barely fermenting? Do you judge just by bubbles from the airlock, or by gravity measurements? The former can be misleading, the latter are reliable. Maybe your fermentation is going well.
If it isn't, you'd better add the new yeast asap. Keep in mind, though, that whatever flavour profile had been produced by the first pack of yeast, it's gonna stay. This way, there will be impact on the end product. Also, whatever stress signals they expressed into your beer, those may also influence the newly added yeast. It's also likely that all oxygen in the beer has been consumed by the first pack, so make sure the new yeast you pitch is healthy and active (I heard that dry yeast already have enough lipids, btw, so they won't need much oxygen - just make sure you rehydrate them in lukewarm sterile water before pitching).
Other than that, yeast has no sex process, so the first pack will not interfere into fermentation character of the second one.