I have a fixed gear bike, Rocky Mountain Boroughs Fixie (see picture). The chain is very tight compared to any free wheel bike. So anytime the tire gets punctured (unfortunately very often) and I want to dismount the wheel, I have to remove the chain by using a chain tool. The question is: does anyone know of some trick to replace the tube without doing all of this?
It depends why the chain is tight.
If there's a chain tug then loosening that is the obvious way. Yes, it's slightly annoying to do that then tighten it up afterwards, but it's easier than the alternative. I use QR axles so for me popping the QR means I can slide the tug off without changing the adjustment and it's all easy from there. But even with a solid axle and nut you should be able to loosen or remove the nut and slide the tug off.
If not, just how tight is the chain? I've never been comfortable having the chain so tight that I can't just slip a tyre lever between the chain and chainring where a front derailleur would go and pedal the chain off that way. Often if the chain is too tight it's an attempt to compensate for bad chain line (ie, there's an offset between the cog and chainring) which is a problem in itself.
Edit: to fix this the only way is to add a link to the chain (so you have more slack), and that probably means using a chain tug to hold the wheel in place. You might not need the tug but unless the cost is critical I would add one. It's about $20 for a cheap one.