Let $\phi : \mathbb{R}^4\to V$ be a field with (complex) target vector space $V$, transforming in a finite-dimensional projective representation $\rho_\text{fin} : \mathrm{SO}(1,3)\to\mathrm{U}(V)$. As it is a field, the representation of the translations $\mathbb{R}^4$ on $V$ is the trivial one, since the field transforms as $\phi(x)\overset{x\mapsto x+a}{\mapsto} \phi(x+a)$. Hence, the field transforms in a finite-dimensional representation $\sigma_\text{fin}$ of the Poincaré group, but the non-trivlal, i.e. interesting, part is the representation of the Lorentz group. Hence, your premise that we "only study finite-dimensional representations of the Lorentz group" is wrong, it's just that the finite-dimensional translations are always represented by their trivial representation.

In the quantum field theory, the field now becomes operator-valued, acting upons ome Hilbert space $\mathcal{H}$. Since the quantum field theory shall have Poincare symmetry, there must be a projective unitary representation $\sigma_\text{U} : \mathbb{SO}(1,3)\ltimes\mathbb{R}^4\to\mathrm{U}(\mathcal{H})$ upon this space of state. By one of the Wightman axioms, we have that $$ \sigma_\text{fin}(\Lambda,a)\phi(\Lambda^{-1} x-a) = \sigma_\text{U}(\Lambda,a)^\dagger \phi(x)\sigma_\text{U}(\Lambda,a)\quad \forall \Lambda\in\mathrm{SO}(1,3),a\in\mathbb{R}^4$$ where on the l.h.s., $\sigma_\text{fin}$ is a finite-dimensional matrix acting upon the vector $(\phi^1,\dots,\phi^{\dim(V)})$, and on the r.h.s., the $\sigma_\text{U}$ are operators on $\mathcal{H}$ are are multiplied multiplied with each component operator $\phi^i$.

We study the finite-dimensional representations because of this relationship - we have to know the finite-dimensional representations to be able to give the "classical" field, and we have to know the infinite-dimensional unitary representation to know how the Poincaré symmetry acts on states, and because the irreducible unitary representations correspond to particles by Wigner's classification. Since the Poincaré group is just as non-compact as the Lorentz group, these are also all infinite-dimensional.

Let $\phi : \mathbb{R}^4\to V$ be a field with (complex) target vector space $V$, transforming in a finite-dimensional projective representation $\rho_\text{fin} : \mathrm{SO}(1,3)\to\mathrm{U}(V)$. As it is a field, the representation of the translations $\mathbb{R}^4$ on $V$ is the trivial one, since the field transforms as $\phi(x)\overset{x\mapsto x+a}{\mapsto} \phi(x+a)$. Hence, the field transforms in a finite-dimensional representation $\sigma_\text{fin}$ of the Poincaré group, but the non-trivial, i.e. interesting, part is the representation of the Lorentz group. Hence, your premise that we "only study finite-dimensional representations of the Lorentz group" is wrong, it's just that the finite-dimensional translations are always represented by their trivial representation.

In the quantum field theory, the field now becomes operator-valued, acting upon some Hilbert space $\mathcal{H}$. Since the quantum field theory shall have Poincare symmetry, there must be a projective unitary representation $\sigma_\text{U} : \mathrm{SO}(1,3)\ltimes\mathbb{R}^4\to\mathrm{U}(\mathcal{H})$ upon this space of state. By one of the Wightman axioms, we have that $$ \sigma_\text{fin}(\Lambda,a)\phi(\Lambda^{-1} x-a) = \sigma_\text{U}(\Lambda,a)^\dagger \phi(x)\sigma_\text{U}(\Lambda,a)\quad \forall \Lambda\in\mathrm{SO}(1,3),a\in\mathbb{R}^4$$ where on the l.h.s., $\sigma_\text{fin}$ is a finite-dimensional matrix acting upon the vector $(\phi^1,\dots,\phi^{\dim(V)})$, and on the r.h.s., the $\sigma_\text{U}$ are operators on $\mathcal{H}$ are multiplied with each component operator $\phi^i$.

We study the finite-dimensional representations because of this relationship - we have to know the finite-dimensional representations to be able to give the "classical" field, and we have to know the infinite-dimensional unitary representation to know how the Poincaré symmetry acts on states, and because the irreducible unitary representations correspond to particles by Wigner's classification. Since the Poincaré group is just as non-compact as the Lorentz group, these are also all infinite-dimensional.