Do computers speed up at higher temperatures?
Lets break your questions into sub-questions:

Faster computer:

The most common measure of computer's "speed" is its maximum clock frequency. This measure has never been an accurate one (Megahertz myth), but it became totally unimportant in recent years after multi-core processors became a standard. In today's computers, the top performance is determined by much more complex factors than just the maximum clock frequency (these factors include both HW and SW aspects).

Temperature's effect on clock frequency:

Said that, we still want to see how a temperature affects the clock frequency of the computer. Well, the answer is that it does not affect it in any appreciable way. The clock for the computer is (usually) derived from a crystal oscillator, which does not heat up at all. This means that oscillator's frequency is independent of the temperature. The signal produced by the oscillator is multiplied in frequency by PLLs. The PLLs' output frequency will not be affected by temperature (assuming that they were designed properly), but the level of noise in PLLs' clock signal will increase with temperature.

The above discussion leads to the following conclusion: the increase in temperature will not increase the frequency of the clock (by any appreciable amount), but can lead to a logical failure due to increased noise in clock signal. 

Temperature's effect on maximum clock frequency:

The temperature has effectively no effect on the pre-defined frequency of the clock. However, maybe higher temperature allows for higher frequencies to be employed?

First of all you need to understand that modern computers do not have their clock rates pushed to the limit of technology. This question has already been asked here. 

The above means that you can increase the frequency of your CPU above the one which was defined by default. However, it turns out that in this case the temperature is the limiting factor, not a benefit. Two reasons for this:


The resistance of the wires increase with temperature
The electromigration rates increase with temperature


The first factor leads to a higher probability of logical failure at high temperatures (incorrect logical values being used). The second factor leads to a higher probability of physical failure at high temperatures (like permanent damage to a conductive wire).

Therefore, the temperature is the limiting factor of processors' maximum frequency. It is the reason why the most abusive overclocking of processors is performed while the processor is super-cooled.

Thermally excited carriers in silicon:

I believe that you were led to the wrong conclusions by the thought that silicon's resistivity reduces with temperature. It is not the case.

While the thermal generation rate is indeed increases with temperature, there is no much use to intrinsic silicon. The fact that the most of silicon used in the industry is doped means that the thermally excited carriers comprise a negligible fraction of the free carriers in silicon; therefore, even huge increase in thermal excitation rates will not affect the density of the free carriers appreciably. Check out this calculator and try to find at which temperatures the density of the thermally generated carriers approaches the usual doping concentrations (\$\geq 10^{16}cm^{-3}\$) - your processor will burn out long before the thermal generation will affect the conductivity of silicon.

Furthermore, the mobility of free carriers tend to decrease with temperature; therefore, instead of the increase in conductivity of silicon, you'll probably observe a decrease which will lead to a higher probability of logical failure.

Conclusion:

Temperature is the main limiting factors of computers' speeds.

Higher temperatures of processors also lead to the higher rates of Global Warming, which is very bad.

Advanced topics for interested readers:

The answers above, to my best knowledge, are completely correct for technologies down to 32nm. However, the picture may be different for Intel's 22nm finFET technology (I found no references for this newest process on the web), and it will certainly change as process technologies continue to scale down. 

The usual approach for comparing the "speed" of transistors implemented using different technologies is to characterize the propagation delay of the minimal size inverter. Since this parameter depends on the driving circuit and the load of the inverter itself, the delay is calculated when few inverters are connected in a closed loop forming a Ring Oscillator.

If the propagation delay is increasing with temperature (slower logic), the device is said to operate in Normal Temperature Dependence Regime. However, depending on device's operating conditions, the propagation delay can decrease with temperature (faster logic), in which case the device is said to operate in Reverse Temperature Dependence Regime.

Even the most basic overview of the factors involved in the transition from Normal to Reverse temperature regimes is beyond the scope of a general answer, and requires pretty deep knowledge of semiconductors physics. This article is the simplest yet complete overview of these factors.

The bottom line of the above article (and other references I found on the web) is that Reverse temperature dependence should not be observed in currently employed technologies (except, maybe, for 22nm finFET, for which I found no data).