"There are many poor people in the country and we have an obligation to support them. So, either you support our welfare system or you have no intention of doing what is right," which fallacy is this?

Identify what fallacy this description would be.

false dilemma

Typical of the “either/or” language of the false dichotomy or dilemma.  

The “leap” in (I hesitate to use this term) logic which careens toward a moral stance suggests an apparent non-sequitur is being employed, as well…thus making the example a compounded fallacy.

I think you may spot two fallacies by noticing that the words 'support' and 'obligation' are ambiguous.

Wow. Talk about a question that is all over the map. Where do I start?

I'll make it short and start with: "There are many poor people in the country." If we were in court, the valid objection would be no foundation. In epistemology (logic), we say, check your premises and definitions. Define "many" and "poor" in the context of what you claim to be true as part of, what I guess is supposed to be, the syllogism.  

Then, as if defined and accepted by all, you say, "and we have an obligation to support them." Who are "we" and why do "we" have an obligation to support them? 

Once we get these definitions and premises agreed to, then we can proceed with the discussion. But I can just a little bit more.

"So, either you support our welfare system or you have no intention of doing what is right," which fallacy is this? This sentence contains the Argument from Intimidation, it is a non sequitur, and also presents the Fallacy of the False Alternative. 

The welfare system uses force to take from some and give to others. This is a moral question of whether it is moral to initiate the use of force against others.

There are multitudes of organizations that take care of the poor. And, they do so without taking from some to give to others. They do it via charity. Therefore, plundering the innocent is not needed to take care of the poor. All of this is not new.

Back in 1850, Frederick Bastiat called the welfare system Legal Plunder. "How is this legal plunder to be identified?" he asked. "Quite simply. See if the law takes from some persons what belongs to them, and gives it to other persons to whom it does not belong. See if the law benefits one citizen at the expense of another by doing what the citizen himself cannot do without committing a crime."

One of my favorite quotes from Bastiat is this: "This question of legal plunder must be settled once and for all, and there are only three ways to settle it:
  1. The few plunder the many.
  2. Everybody plunders everybody.
  3. Nobody plunders anybody."