First, it depends on your broker. Full service firms will tear you a new one, discount brokers may charge ~nothing. You'll have to check with your broker on assignment fees.
Theoretically, this is the case of the opposite of my answer in this question: Are underlying assets supposed to be sold/bought immediately after being bought/sold in call/put option?
Your trading strategy/reasoning for your covered call notwithstanding, in your case, as an option writer covering in the money calls, you want to hold and pray that your option expires worthless. As I said in the other answer, there is always a theoretical premium of option price + exercise price to underlying prices, no matter how slight, right up until expiration, so on that basis, it doesn't pay to close out the option.
However, there's a reality that I didn't mention in the other answer: if it's a deep in the money option, you can actually put a bid < stock price - exercise price - trade fee and hope for the best since the market makers rarely bid above stock price - exercise price for illiquid options, but it's unlikely that you'll beat the market makers + hft. They're systems are too fast.
I know the philly exchange allows you to put in implied volatility orders, but they're expensive, and I couldn't tell you if a broker/exchange allows for dynamic orders with the equation I specified above, but it may be worth a shot to check out; however, it's unlikely that such a low order would ever be filled since you'll at best be lined up with the market makers, and it would require a big player dumping all its' holdings at once to get to your order.
If you're doing a traditional, true-blue covered call, there's absolutely nothing wrong being assigned except for the tax implications. When your counterparty calls away your underlyings, it is a sell for tax purposes.
If you're not covering with the underlying but with a more complex spread, things could get hairy for you real quick if someone were to exercise on you, but that's always a risk. If your broker is extremely strict, they may close the rest of your spread for you at the offer. In illiquid markets, that would be a huge percentage loss considering the wide bid/ask spreads.