Is there anything like "positive suffering" for Buddhists? In some religions, suffering is seen as a positive thing, some people even see it as a necessary step for growing, however in Buddhism suffering is seen as something that should be avoid or eliminated, we don't invite suffering, we want peace and there is nothing wrong with it.

So, is there anything like a positive and necessary suffering in Buddhism?
Yes, there is, as mentioned in the Upanisa Sutta of the Samyutta Nikaya:

&quot;Faith, monks, also has a supporting condition, I say, it does not lack a supporting condition. And what is the supporting condition for faith? 'Suffering' should be the reply.
(SN 12.23, Bodhi, trans)

Bhikkhu Bodhi wrote an article on this sutta here:
http://www.accesstoinsight.org/lib/authors/bodhi/wheel277.html
As he puts it:

The Buddha's declaration that suffering is the supporting condition for faith points to the essential backdrop to the awakening of the religious consciousness. It reveals that spiritual awareness and the quest for enlightenment do not arise spontaneously in harmony with our natural modes of world-engagement, but require a turn &quot;against the current&quot; a break away from our instinctual urges for expansion and enjoyment and the embarkation in a different direction. This break is precipitated by the encounter with suffering. Suffering spurs the awakening of the religious consciousness in that it is the experience of suffering which first tears us out of our blind absorption in the immediacy of temporal being and sets us in search of a way to its transcendence. Whether in the form of pain, frustration, or distress, suffering reveals the basic insecurity of the human condition, shattering our naive optimism and unquestioned trust in the goodness of the given order of things. It throws before our awareness, in a way we cannot evade, the vast gulf stretching between our ingrained expectations and the possibilities for their fulfillment in a world never fully susceptible to domination by our wills. It makes us call into question our schemes of values built upon the bedrock of personal expedience. It leads to a revaluation of all values and a new scale of worth indifferent to the claims of self-concern. And it opens us to confidence in an unseen order of relations and inter-connections, an order in which the values that emerge, so often in forceful opposition to the old, will find their proper justification and reward.

I guess I'd put it more simply: suffering leads one to put one's faith in spiritual practice because one no longer sees pleasure in mundane existence. It leads one to gain faith in the Buddha's teaching specifically because it teaches that suffering is a fact of life, which one has come to verify.
So yeah, suffering is important.  There is another sutta (Padota Sutta) that talks about four types of thoroughbred horses as a simile for humans:

&quot;Once more, monks, here we may have a goodly thoroughbred man who neither hears it said... nor yet with his own eyes beholds... nor is his own kinsman or blood-relation afflicted or dead, but he himself is stricken with painful bodily feelings, grievous, sharp, racking, distracting, discomforting, that drain the life away. Thereat he is stirred, he feels agitation. Being so stirred he strictly applied himself. Thus applied he both realizes in his own person the supreme truth, and sees it by penetrating it with wisdom. Just as, monks, that goodly thoroughbred steed on being pierced to the very bone is stirred, feels agitation, even so using this figure do I speak of this goodly thoroughbred man. Of such a sort, monks, is the goodly thoroughbred man in this case.
(AN 4.113, Woodward, trans)

So there is definitely the idea that suffering spurs one towards enlightenment, at least in a preliminary way.