This can be an issue, but the first thing to determine is whether you’ll be printing digitally or traditionally. Because, in digital printing, you can have all the colors you want for the same price. Traditionally, the printer will charge for mixing up each color, applying them, and also doing the pre-printing work like making “screens” (basically a template for where each color would go). So, traditional printing costs more as you add colors.
Another (sort of) interesting fact is that most traditional printing presses, at least in the US, are 2-color. So, to print a 3-color job, the printer has to print the first 2 colors on your job, and then let it dry a bit, and then reset the press, realign things, and send the whole job through again. There’s inevitably more spoilage due to misalignment, and it takes more time, and is (I’ve heard) sort of a pain!
So, cost-wise, for traditional printing (here in the US), you can basically double the price of the average 2-color job to get the 3-color equivalent. Yikes!
If you do choose to design a 3- or more color logo, I’d suggest developing a 1- or 2- color logo in case you need to use it to minimize printing costs in the future.