Why does WPA-PSK not use Diffie-Hellman key exchange? Is there a reason, why in pre-shared key mode, Wi-Fi session key is not secured with something like Diffie-Hellman but instead is derived from PSK key and some information exchanged in clear during sign-on?
As Thomas pointed out in his comment, in WPA-PSK the parties already have a shared secret (the pre-shared key), so there is no need to go through the step of Diffie-Hellman key-exchange -- unless you want perfect forward secrecy (PFS) (see update below). 

In WPA, the client and the access point goes through a step known as the "4-way-handshake" to verify that the other party is in possession of the pre-shared key and to derive session keys for actual (encryption/authentication) usage. This step involves nonces from both sides, and some cryptographic computations, with the end result that 1) both parties can verify that the other party is in possession of the common secret and 2) they both have derived a shared session key which they will use for the actual encryption.

The picture below shows the steps in this process. The Supplicant is the client and the Authenticator is the access point. The PMK is the shared secret. In a typical home-user setup, the PMK is usually derived from a password using PBKDF (this password is what you normally punch in when connecting to a wireless network). The PTK is the derived key that they end up actually using for encryption.



UPDATE

As mentioned in the comments, the above does not actually answer why WPA-PSK does not use DH. However, before discussing that, let's clarify what DH has to offer over what's currently being used, and also why the use of pre-shared keys diminishes some of those advantages. 
I claim that the one big advantage of using DH is that if offers perfect forward secrecy (PFS), while the ability to "separate" users from each other on the same WLAN is more debatable. 

First we need to be clear about the kind of attacks we want -- and can -- be able to protect against. 


Outsiders, i.e. attackers not knowing the PSK: If the PSK contains sufficient entropy (either drawn completely at random from a big key space or derived from a sufficiently long/complex password) and the PSK is kept safe and never leaked then DH does not offer anything over the current solution. 
If, however, the PSK is either lost or derived from a weak password, then the current system is completely broken. The reason is that the session keys are completely derived from the PSK, hence anyone who knows this can decrypt all traffic. Even worse, since it does not offer PFS, an attacker can go back an decrypt all previous traffic as well. The use of DH would prevent this. Since in practice most home WLANs uses a weak password to derive the PSK, DH would indeed be an upgrade.
Insiders, i.e. attackers knowing the PSK: Now consider the scenario where the attacker knows the PSK. What could the handshake offer then? As mentioned above, with the current system the answer is nothing.  So what about DH? Assuming that in this scenario the PSK is only used to authenticate the DH exchange (say with a MAC), the DH would still offer security against a passive attacker, while nothing against an active attacker. Why? In the passive scenario, the attacker has to break the DH exchange which we assume is intractable. On the other hand, in the active scenario, the attacker can do a classic MitM attack, since he can forge valid handshake messages. Thus, under the typical conservative approach taken in formal security models, we would conclude that DH offers no security benefits over the current scheme since we always assume that the attacker is active. However, in practice, conducting an active MitM attack can sometimes be tricky to pull of. Thus, using DH significantly increase the cost and difficulty for the attacker. Since it forces him to be actively present during the hand shake, this also prevents large-scale passive snooping.


So to conclude, using DH would provide huge benefits in the sense of providing PFS. Also, it also provides some degree of protection of insider attacks, as long as the attacker stays passive.  

So why was the 4WHS not designed to use DH?

Full disclosure:
this will just be speculation on my part, since I do not know the arguments that went in to the design decisions of the IEEE 802.1i standard. 

I guess that at the time WPA (or more correctly IEEE 802.11 RSNA) was designed, the value of PFS was probably not fully appreciated. In hindsight of the NSA revelations, it is of course easy for us to understand its importance now, but at that time it might not have been so clear. Thus, if you are not required to have PFS, and you already have a PSK available to your protocol, then there is a very simple reason why DH would not be used: efficiency. Symmetric primitives are much much more efficient than public-key primitives, hence if you can get away with only using that, the choice is obvious. 

*Incidentally, the IEEE 802.11 standard actually do support a DH-based key-exchange built on a password-based AKE called Dragonfly. However, this variant is only used when the network is in ad-hoc mode.