I'm another in the "it depends" camp. What kind of tired are we talking about? I'm chronically sleep-deprived and just not a high-energy person overall, so I'm pretty much always tired. In my normal state of tiredness, yes, I will work out at any level.
OTOH, if we're talking a lengthy period of chronic stress, exhaustion, and overwork (or recovering from illness)... then, no. I take it easy. I have to. When I was younger I would just power through, but (fortunately? LOL) I can't do that anymore.
On a semi-related note, I'm really starting to realize how important
scheduled rests are, for my overall health. I'm self-employed, and when I'm under contract my stress and busyness are pretty intense--but limited, only lasting for a few months at a time. The last few years, though, I've been entirely freelance, so I'm working all the time--or if I'm not working, I feel like I *should* be working. My body and mind really MISS those built-in rest periods, where I
know I'm off the clock and can truly turn off. Someone recently posted about the value of taking scheduled rest weeks from working out, and it's the same thing. There is really something to be said about INTENTIONAL days off, and not just missing workouts here and there.
It's a tricky balance, to be sure. But what isn't??