If a man dies with no Will, should his children continue to pay the mortgage on his home?

If there is no surviving spouse as your question implies, the children will stand to inherit the house. However, the lender may foreclose if the loan is not paid. Therefore, the question really is does one or more of the children want the home--which also subsumes the question of is there enough equity in the home that it's worth paying the loan so as to then be able to sell the home and profit by it? If the answer is yes--someone wants the home for personal reasons and/or the home is valuable has substantial equity--the children should contact the lender to discuss payment arrangments.
However, no one can be made to inherit anything--if the children don't want the home, or the loan is so large that it's not worth getting the home, then they and the estate i.e. via the court-appointed administrator can refuse to pay and let the lender get the home. There will be no consequences to them if they disclaimed the inheritance of the home.
IRS debt it becomes a debt of the estate, but not of the heirs. The IRS can try collect from the money and/or assets in the estate, before they are distributed.