NP-Hardness of a special case of orthogonal packing problem
I think that the paper by Klaus Jansen and Roberto Solis-Oba "An OPT + 1 Algorithm for the Cutting Stock Problem with Constant Number of Object Lengths" has a partial answer to your question. They consider a special case of your problem known as Cutting Stock problem when the number of diﬀerent object types is constant and defined as follows:


  In the cutting stock problem we are given a set $T = \{T_1,T_2,\dots,T_d\}$ of object types, where objects of type $T_i$ have positive integer length $p_i$. Given an inﬁnite set of bins, each of integer capacity $\beta$, the problem is to pack a set $\mathcal{O}$ of $n$ objects into the minimum possible number of bins in such a way that the capacity of the bins is not exceeded; in set $\mathcal{O}$ there are $n_i$ objects of type $T_i$, for all $i =1,\dots,d$.


The authors claim that


  it is not known whether the cutting stock problem can be solved in polynomial time for every ﬁxed value $d$.


And they propose $OPT+1$ approximation polynomial-time algorithm when $d$ is fixed.

Since it's not proved that this special case is in $P$, this is the evidence that your problem is $NP$-hard.

Addendum: it's known that the case with two object types ($d=2$) is polynomially solvable, but for $d=3$ there is known only $OPT+1$-approximation.