Do debit cards have interchange fees?

Yes, debit cards have interchange rates, just like a credit card. However, debit cards don't carry the same back-end costs for issuing banks that credit cards do. They don't have perks and rewards for their customers, so issuing banks don't have to pay hefty costs for those. Nor do they typically get defaulted upon - customers usually only spend what they have. So, since these cards are "low risk" for issuing banks, they typically have far lower interchange rates than their credit counter-parts.

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In addition, many debit cards are "regulated" by the federal government. In the midst of the 2008 financial crisis, new legislation was proposed and subsequently passed by senate. It was called the Durbin Amendment. The intent was to create some financial regulations to protect consumers and the banking industry from another potential banking crisis. One of the large side-effects was that interchange rates were "capped" on all debit cards issued by big banks. "Big Bank" was defined as any bank with over 10 billion dollars in assets, so think Bank of America, Citi,  Wells Fargo, US Bank, etc. The interchange rates for any debit card they issue  would have a cap of 0.05% and 22¢ per sale. 

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Due to the fact that about 80% of the debit cards issued in the US come from a big bank... well, you can see that you'll likely be receiving a lot of regulated cards! As such, it's relatively easy to guess what the underlying rate is when you receive a debit card in person, it's likely the aforementioned 0.05% + 22¢.

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Smaller regional/community banks and credit unions still have interchange rates on their debit cards, too. Usually, they're around 1.00% (see the Visa/MasterCard rates here). So, more expensive than a regulated debit card, but still a lot cheaper than a standard credit card.