Yes, it will cause wear on the cassette such that a new chain will not be able to seat right on the cogs (and so your shifting and skipping issues won't improve). The next thing to go with a worn chain would be the chainrings. It's much cheaper to just replace the chain often, and if you have a really good chain (or are easy on them), you may get a ton of miles out of it before the chain wear indicator tells you to get a new one and it starts skipping. In that case, you may need a new cassette even if you don't wait until you get bad shifting or skipping problems just because of all the use. It might depend on how quickly you wear out chains and the quality or durability of the components (the most expensive cassettes don't necessarily last the longest).