So lets say we are currently at block 240,000. If I were to say I will use the hash of block 240,010 as the seed to a random number generator (or some number of bits from the hash) for a number guessing game, would this be a secure mechanism for seeding the RNG?
The only attack I can think of is someone with an interest in the outcome of the RNG who has enough computing power to solve block 240,010, check if the result favours them, if so submit it to the network, otherwise discard it.
Is this a feasible attack? Is there any other issues?
The attack is feasible under many reasonable definitions of an adversary. There's a good paper on that by Pierrot/Wesolowski:
http://eprint.iacr.org/2016/370.pdf. Shortly, if you're defining a single random bit from a block, then in case like yours a miner with 25% of mining power can take a desirable outcome probability from 50% to 74% by spending ~1 block reward(>=25 BTC atm).
Better schemes of randomness extraction from the Bitcoin blockchain could be proposed, but first you need to define your use-case and goals precisely.