Yes. If you wouldn't drink it, don't cook with it.
Whatever it tastes like out of the bottle, it will add that to the dish. Cook with a wine you might pair with the dish (light wines for seafood, chicken; heavier wines for meats and stews).
Don't use a fruity wine unless you want your dish to have some fruit notes. Don't use a very dry wine if you're making a sweeter dish.
Speaking from personal experience mulling wine many times:
In general, you want a dry or semidry red, of innocuous flavor profile. You do not want anything with strong tannic, acidic, alcoholic, brett or sweet flavors; these will become concentrated while mulling and quite unpleasant. Your ideal mulling wine is an inexpensive, inoffensive, young merlot, burgundy, petite syrah, tempranillo, beaujolais, or other "middle-of-the-road" wine, maybe slightly on the sweet side. Cabernet sauvignon, chianti, rioja, and similar wines tend to be poor choices, although of course it depends on the individual wine. Also, look out for high-sulfite-added wine which also can develop off flavors.
I'll contradict Sarge here and say that you do not want a wine which is turning towards vinegar; you'll end up with a very sour crock pot full of mulled vinegar. However, mulling is an excellent thing to do with wines which have been oxidized (but not vinegared) and lost a lot of their flavor, either through being open too long or too long on the shelf. Certainly if you spend more than $9 a bottle in the USA for wine for mulling, you've made a mistake.
This is very similar to how you would choose a sangria wine. The main difference is that for sangria you want bright and acidic flavors, whereas for mulled wine you want heavier, darker flavors.
Best Answer
I don't think you will get any off taste, but it will change the nature of your drink - at the minimum, it will be much less alcoholic afterwards, likely far more concentrated, and possibly more spiced (depending on whether straining the spices out is something you would think to do before this long simmer)
Looking at your recipe, it looks like a mulled wine flavored syrup added to wine - the actual quantity of wine is supposed to only be heated briefly to prevent the alcohol from evaporating out. The recipe actually mentions this as a reason why the spices are infused into a syrup first, instead of the whole quantity of wine - the boiling would evaporate quite a bit of alcohol out of the wine, I think about 25% would be lost if you simmered for an hour.
If you still wanted to keep your mulled wine hot for an hour or two, I would suggest you look for mulled wine recipes that do have extended heating of the wine (maybeso something like this, though it's just the first I saw), the better to balance the spices so they don't over-infuse on you on one hand, or have the wine over-concentrate them. If you want your mulled wine to still be strongly alcoholic, you might add some spirits at the end, just before serving to balance the loss.
So, yeah, the summery is, the wine will likely not be spoiled, or develop off flavors, just by being heated for extended periods of time, this happens in recipes, including mulled wine recipes, often enough for serious problems to be unlikely - though it will change your recipe's effects. The loss of alcohol is a factor, but it can be compensated for. The level of the spices might be tricky to manage - you might want less of you're going to be simmering them in the wine the whole time, you might get less out of them (ie, need more) if you only simmered them in the wine and skipped the hard boiling syrup stage (it might balance out?), or you might simply strain them out of the syrup before adding the rest of the wine.
Of course, you might find it easier to prepare the syrup in advance, and add to the wine and heat them both just before serving - this would make the recipe quick to assemble and sidestep some of the complexities that simmering that long might cause. One of the comments on the recipe even suggested this might add a touch of complexity to the drink, since the flavors would have a bit more time to meld together.