What would be the effect of adding small (~<10mm) lengths of conductor between a CPU's contacts and socket
Your CPU has several types of pins/signals:


Slow signals like SMBUS, etc.


These wouldn't care.


Fast signals like PCI-Express, RAM, etc.


These all use transmission lines, single ended or differential. Any impedance discontinuity in the transmission line would create signal reflections and corrupt signal integrity, so you would need to make a riser connector with controlled impedance. This can be done (and it is  done) but it will involve tiny high precision machined parts and obscene prices because this is a very low-volume, specialty custom product. Not to mention how crosstalk between nearby signals would be handled. Perhaps tiny coax pogo pins? Add an extra zero to the price.


Power/Ground


These require extremely low inductance which is achieved by having literally tons of pins. Here's a random PC CPU pinout from the internets:



Enlarge it and look at all the supply and ground pins (anything that starts with "V"). More than half the pins are supply and ground, and this  adds to the cost. 

Adding inductance (ie, length) would degrade power integrity. 

These are the reasons why motherboard CPU sockets sit as low as possible on the board and keep the connections as short as possible...

So i would strongly recommend you find a cooling solution that doesn't require this. It is a lot easier to move heat up by a few mm (a 10 cents copper shim will do the job).