The key word in your question is "should". So you are asking us to make a moral judgement about the value of dissent in society and whether a government should put up with it.
Should they? Absolutely. One of the foundations of democratic government, what makes it functional, is compromise. That is, compromise between opposing ideas and elements of government. What dissent does is offer (hopefully) constructive criticism of government policies and actions. If there is enough dissent and it is directed in the right places, it can improve government and society. Government reacts to it, because in the end, it is accountable to the public.
If you want to be more objective still, simply compare the countries and governments that do and don't tolerate dissent. Do: The US, Japan, Britain, Germany, France. Don't: Iran, Pakistan, Myanmar, Cuba.
Which list would you like your government to belong to?
I think it depends not so much on what the dissent groups motivations are but what their actions are in expressing those motivations. One of the freedoms guaranteed to us under the Bill of Rights in the United States (a freedom that does not exist in some other countries) is the right to free speech and free expression. This includes the freedom to voice a dissenting opinion. The key here, however, is the freedom to VOICE that opinion. It is fine to speak about dissent, to rally others to dissent along with you, to march, to wave flags, to advertise from billboards and airwaves and, most importantly, to rally for any change that you believe in through voting and encouraging others to vote.
However, when a dissent group becomes a hate group and their actions turn to violence, that is an entirely different story. Bombing an abortion clinic, for instance, because you believe abortion is morally wrong is taking dissent to an unacceptable level. Attacking people because of their race, religion, gender or sexual orientation is also a form of dissent that should not be tolerated by a government whose main goal is to protect and serve the population.
I would say that if the government believes in free expression, then actively encouraging the presence and articulation of dissent groups is vitally important. A political order that stresses the idea of a free expression is one that must be able to cradle and seek out dissenting voices. This helps to enhance the vibrant tapestry of political expression and ensures that all, or as many as possible, are heard in the process of deliberation and execution of political forms of justice. The challenge would be if particular dissent groups move into a realm where action is counter productive to the sensibilities of a great democracy. This might be where limitation is warranted and is challenging because drawing and adhering to that line without denying expression of dissent is complex, at best.
I suppose this depends to some extent on what kind of dissent groups you are talking about. But in general, as long as dissent is only verbal, it is completely essential that government should tolerate these groups.
The two underlying values I think you have to consider are these.
First, on the side of not allowing dissent, is the value of stability. We all want a stable society where law and order are upheld.
Second, on the side of allowing it, is the value of freedom. Our country is founded on the idea that all people should be as free as possible.
To me, you have to decide between these two. Do you want a free society or do you want one that is completely orderly? I know which I prefer.
Those actions of dissent groups which are not criminal in nature, should be tolerated by government. So underlying value, whether or not a criminal act.