Good news – the IRS cannot chase you forever for your unpaid taxes but only if you filed an “honest” tax return. Federal law limits the time they can pursue you to ten years after the date of assessment, unless you have filed for bankruptcy or have submitted an offer in compromise.
More good news – the IRS has three years from April 15th to examine your return and determine whether you owe more money. The precise rules involved can get complicated and are beyond the scope of this short treatment of the statutes of limitations. Once the three years have expired, they cannot assess a deficiency against you. There are several exceptions to this rule so if you feel they have violated the time limitation seek professional counsel before you assert this defense.
Some bad news – If you do not file your returns for a given year the statutory period in which they can pursue you never starts, and therefore never ends. They can chase you forever.