Americans living abroad should still be required to file their income taxes. They still benefit from American services such as social security. Just because they happen to be living abroad right now does not mean that they will never return. They should be required to support their country wherever they happen to live.
Are they making their money here? If they are making an income from here, then yes I think so. Most of my income comes from overseas, and I still pay taxes in. If they are making their money else wear and not from here then no, I don't think they should have to. It really depends on the situation
If they company the ex-pat is working for is based in the country they are currently living in, I don't understand why they would have to pay taxes to the US.
However, if they are an employee of a company based in the US, regardless of where they are currently choosing to live, then yes they should pay taxes just like any other US employee.
For American citizens living and working in the US, paying taxes is part of a large-scale trade-off: citizens have access to certain government services, and more generally enjoy government support of key industries and infrastructure, that help keep the economy and the nation going. In return, citizens pay taxes that help fund government operation. For citizens living abroad, however, they have access to far fewer of these services, and indeed, in most cases they do not require these services. Hence, they should not have to pay taxes, or at most, they should have to pay a much lower tax.