There isn't a time limit, but taxes will be owed if the collection appreciates in value between the time it is inherited and the time it is sold.
1. You have a taxable capital gain whenever you sell inherited property for more than the fair market value on the date the prior owner died.
2. If you are audited and can't prove the FMV, the IRS is allowed to asses tax on the entire sales price. So you should get a qualified, signed appraisal as soon as you can, if you intend to keep the collection.
3. After getting an appraisal, any tax owed depends on whether the specific coins go up or down in value after that. Coin pricing is complicated and there is no way to really guess if, or how fast, prices will change. So any capital gains tax you will owe is hard to predict.
4. When you do sell the coins, if you have a taxable capital gain, you should make an estimated tax payment by the end of the tax quarter. That means: