Not necessarily. As long as the debt is invested well, it is absolutly fine. If the debt is used to increase the size of the economy, that means more taxes and hence the debt can pay itself. The other scenario is the government investing into something, which increases in value. Keep in mind that the governments rates are lower the private rates, so if a government builds a lot of housing for its citizens, that increases debt, but is still cheaper then everybody building their own housing themself. Similar effects can be had by buying companies.
Debt is a tool, it can be used for both good and bad. It really depends on how good the government is.
MMT would claim rather simply that the debt can be paid back by printing more money. More money in the system means more people are employed. If you have full employment, more money leads to inflation and to fight inflation governments need to raise taxes, to take money out of the system. Obviously taxes can not be raised to infinity.
So no MMT does not claim the US can just borrow money without long term costs.