The cases where using a higher value resistor will damage a circuit exist, but are a bit less usual than the cases where it may simply produce a weaker result than desired, or a different frequency response than desired.
The primary situations where you do have to watch out come about where "feedback" is used to adjust one thing in relation to something else, and so putting a larger resistor on the "sense" side of the circuit can cause the "output" to be too high. Examples you might run into would be adjustable voltage regulators (obvious potential for damage with wrong output) and op-amps (higher than desired output, though more often distortion than damage).
It seems quite likely the the role of the resistor in your circuit is to limit the current through the LED - protecting that (and to a more vaguely best-practices degree, the ATmega pin) from damage.
For your needs it could be worth noting that a parallel combination of two 510 ohm resistors will give an effective resistance of half that, or 255 ohms. This is very close to the design intent, so might be a decent substitution (cheap resistors have a larger tolerance for piece-to-piece variation of value than that anyway).
Normally I would say the desired 270 ohms feels a little small, and that I'd prefer to start with 330 ohms. But if you do parallel two 510's, the 255 ohms would give just under 20ma of current if connected directaly across a 5v supply. Given your LED will have some forward voltage drop (varies widely with color), and your Arduino pin a little of its own, the actual voltage and thus current will be less - as a guess the end result is likely to be in the 15ma range, and not unreasonable for many common LEDs.
Resistors would also be a fairly easy component to salvage out of old electronics - the leads will end up short, but other than that you basically just need to read the color code, and perhaps confirm the value after removal with a meter.
Ultimately though it is worth buying an assortment of resistors - the 1/4 watt size are generally preferred for breadboarding as the leads are the best fit.