Will the IRS forgive any tax debt if requested?

There are some ways to do so:
You can opt for 'Life Jacket' if you're underemployed or unemployed. Though this process the IRS freezes the collection account and refers it to as uncollectible status.
Another is 'Cents on the Dollar' in which you pay 10 cents or less on each dollar and settle it.
If you file bankruptcy, you can get a discharge from your tax debt.

If you owe the IRS but you are unable to pay them off, you may qualify for the IRS debt forgiveness. However, your current financial condition will get the primary importance. The IRS can't collect more than you can afford. If you can afford a partial amount of tax debt, you can apply for the IRS government payment plan Offer in Compromise (OIC) to resolve the rest of the amount. Once you can get approval for a reduced amount, you can repay the amount in a lump sum or in fixed monthly instalments.

Another way to do so is "Wage-Earner's repayment plan". A Chapter 13 bankruptcy allows a taxpayer to enter into a repayment plan in accordance with his ability. But you need to show the IRS that they will get more by accepting your offer than they will by forcing you into bankruptcy.