I think it depends a lot on the couple, the circumstances, and the setting. In some places, interracial couples still raise eyebrows (although from my experience this is less and less the case). But, especially if you're in a more urban area with a racially mixed population, I don't think it's as important to most people as it would have been in the past. It also depends on what your plot focuses on overall. If it's about the struggles of the couple, or meeting each other's families, etc, the issue of race could factor in as another obstacle to their relationship. If it's about, say, trying to run a marathon to win back your ex, I don't see any real reason to focus on race as it doesn't play a role in the characters' motivations or problems.
So, for example, my best friend is white and dating someone who is Asian (granted, they are a gay couple, so that's where more issue comes in). In their day-to-day life and among friends, race isn't an issue other than my friend occasionally commenting on how often they use their rice cooker. When it comes to dealing with each other's families, though, a lot more comes out. His boyfriend's family is very traditional so there is some culture clash there. Still, the issues aren't strictly about race, but about the different traditions they were raised in. On the other hand, when I once dated an Afrikaner, he was consistently amazed that my roommate was black since he'd been raised in a place where that would not be the norm. And, on the far end of the spectrum, a friend once got upset watching Bend it Like Beckam because, while her family had no problem with her traveling thousands of miles and rarely seeing them to go to college in the States, they would never forgive her if she dated a white guy even if he did play cricket. Basically, some individuals and cultures make a bigger deal of interracial dating than others. Someone who has lived their whole life in urban America probably wouldn't think twice, while traditional Sikh parents from Calcutta could throw a fit.
So I'd say make race an issue only if you want it to be a major point in the plot and the conditions warrant it. If you want to include a few references just for realism but don't want it to be a big deal, a couple of off-hand comments could do it (don't know what races your characters are, but the example of grocery shopping with my friend and him saying, only half-joking, "And, of course, I have to buy more rice because I'm living with an Asian man," comes to mind). But, generally, I don't think race has to be an issue unless the setting and characters make it one.
"How do you know I'm mad?" said Alice.
"You must be," said the Cat, "or you wouldn't have come here."
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