Why Lorentz group for fields and Poincaré group for particles? Wigner treatment associates to particles the irreps of the universal covering of the Poincaré group
$$\mathbb{R}(1,3)\rtimes SL(2,\mathbb{C}).$$ 


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*Why don't we consider finite dimensional representations of this group? 


I understand we ask for unitarity when representing its action on states, so the representation cannot be finite dimensional. However we do consider finite dimensional representations of the Lorentz group $O(1,3)$ and associate them to fields. 


*Why associate the Lorentz group to fields? 

*Why do we look in this case for finite dimensional representations? 

*What do we associate to unitary representations of the Lorentz group?
 A: 1) Why don't we consider finite dimensional representations of this group?
As you said, we ask (anti)unitarity, so it is impossible to find finite-dimensional representation.
2) Why associate the Lorentz group to fields?
The essence of the answer is what Trimok already said in his comment: the "translational part" of the Poincarè group is already represented by the argument of the field. That is for a general multi-component field you postulate the transformation law, for each element $(a,\Lambda)$ of the Poincarè group, given by
$ \psi'(x') = S(\Lambda) \psi(x)$
where $x' = \Lambda x + a$.
It seems a natural request for the transformation rule of a field, think of the non-relativistic case of the Schroedinger field: you expect that the operator creating a particle in position x with spin m=1 is seen as the operator creating a particle in position x+a with spin m=1 by an observer translated with respect to you. Spin, or any "inner" part of the field, should not be afflicted by translations.
I don't know if there is a more deep or rigorous explanation for this.
So you see that fields are distinguished by $S(\Lambda)$, and  so only the Lorentz group is relevant for this purpose.
Note that no general request is asked to $S(\Lambda)$, a part for being a representation (I don't remember if it is allowed to be a projective representation though) of the Lorentz group. 
In the case of the Dirac field, in order to pin down the explicit form of $S(\Lambda)$, it is made another request, that is it leaves invariant the form of the Dirac equation. In the end it turns out it must not necessarily be unitary.
