Unfortunately, you cannot get a refund.  

You can elect to report the COBRA payments as a medical expense and potentially get a deduction if you are able to Itemize your deductions.  

Medical expenses are deductible if they exceed 10% of your Adjusted Gross Income (AGI).

You can enter these medical deductions by going to the Federal Section.

1. Select "Deductions & Credits"
2. Select "Medical"
3. Proceed to enter your medical expenses including the premiums you paid with COBRA for medical insurance.

Flexible Spending Accounts (FSA)s are usually pretax dollar contributions and therefore do not receive any sort of deduction for any medical expenses paid out of them. 

What is an FSA?

What's the purpose of having an FSA while on COBRA if you cant deduct from your gross income? What's even worse is the fact that no one informs you ahead of time that the COBRA FSA is going to be completely useless for most people.

You can read about a COBRA FSA here.

It seems to me that having a COBRA FSA wouldn't make sense unless you had contributed most of the money while still employed, but had not spend much of it, because the COBRA FSA lets you continue to spend the amount already contributed (i.e., the pre-tax amount). To answer your question ("What's the purpose of having an FSA while on COBRA if you cant deduct from your gross income? "), this is why you might want to do it, rather than forfeit the money you already contributed.

Otherwise, I don't see why you would want to do it. A good HR department would walk you through this as part of the exit process.