Digital receivers for the reception of non pulse shaped signals We are used to deal with pulse shaped signals, for example signals that go through root raised cosine filters at the transmit, and then  matched with root raised cosine filters at the receive

In addition to these functions there are carrier phase and frequency synchronization, symbol timing synchronization at the output of the matched filter , equalization,  etc..

Very briefly please, how would things be different at the receiver, if any, when receiving a signal that has not been pulse shaped by any conventional pulse shaping filter at the transmit(for whatever reason)?
Since you are talking about digital communication you are going to have a symbols.
When u send a symbol $s(t)$ you are going to send it with a symbol time $T_{sym}$
so we can say that any symbol that is sent equals to $s(t)\Pi(\dfrac{t}{2T_{sym}})$ where $\Pi$ is the rectangular function.
So basically any discrete time signal has a rectangular shaping pulse.

E.g.

In BPSK you will be sending one of the following symbols:
$$
    S_1(t)=1
$$
$$
    S_0(t)=-1
$$
Since I am sending the symbols for a known time $T$

I am practically multiplying them with a rectangular window with the length of $T$

so for the data series $a$ the signal $S(t)$ will be :
$$
   S(t) =  \sum_{k=0}^\infty S_{a_k}(t)\Pi(\dfrac{t-kT_{sym}}{ 2T_{sym} })
$$

And the matched filter in the receiver will be the average over $T_{sym}$ (or integration over $T_{sym}$) which is otherwise known as a rect.

Which is the same way you would do with other shaping pulses