Does it make sense to say yes to intimacy sometimes even when I don't feel like it? I want to keep the communication in my marriage strong and make sure we're maintaining a feeling of connection, but I also don't want to feel like I'm compromising.
Intimacy of both the physical and emotional varieties are essential to any successful marriage or long-term partnership. Ideally, they would complement each other and build off each other: being close physically opens you up to share yourself emotionally, and sharing your deepest thoughts in a way that makes you feel heard and understood and appreciated leads the way to attraction and desire. The interplay of the physical and emotional is what separates a "real" relationship from, say, a one-night stand or a friends with benefits situation or even a long-term partnership that isn't going that well.
There are lots of reasons to say no to intimacy of either variety, but it's worth overcoming the exhaustion or the self-consciousness or whatever it is in order to maintain both kinds of connections with your spouse. You'll be better off finding a way genuinely to overcome those feelings, though, rather than saying yes in spite of them. Your partner may appreciate the chance to be intimate regardless, but your relationship will benefit more overall if you actually get excited about what you're doing. Still, if you go for it even if you're not that interested, your body may take over and help you get interested anyway. Also, if you ever went to Jewish summer camp, you've heard the phrase "double mitzvah" in relation to having sex on Shabbat, so perhaps you could set a goal to get yourself in the mood on Friday nights. (Also, googling "double mitzvah" got me to a great blog post  about "double mitzvahs.")
Give yourself permission to talk to your spouse about your concerns. Say that you really value your physical and emotional closeness, but you're not always feeling it at the same time, and you hope you can be open and honest with each other about your needs. Guess what? That's called emotional intimacy! Feeling stressed? Ask for a backrub. Maybe it will lead to more, but it could also be a valuable way to connect physically that doesn't come with a broader set of expectations.
If you feel like you're compromising by having sex with your spouse, then there are likely larger communication issues going on that you'd both benefit from addressing. I'm a huge advocate of couples counseling, and even if your first instinct is to say, "We're fine. That's not for us," give it a second thought. An outside perspective can provide insights into your relationship that you'd never have found on your own and can give you some newfound freedom to get to know your partner on fresh terms.
Also, when you're even a little bit in the mood, consider initiating the intimacy yourself so that you're not always the one who's in the position of saying yes or no. If your spouse says yes, then, great! You'll get to connect, you'll both have fun, your spouse will likely appreciate the change in dynamic and you'll likely feel empowered by it as well. If your spouse says no, you'll see what it feels like to be on the other side of that experience, which may change the way you respond next time.