A proper cup of tea requires freshly boiled water to stew the tea leaves.  A cold cup will sap some of the heat from the water, so you might be brewing at 98 degrees C instead of 100 deg C, for example.  The hotter your mug, the less heat will be sapped from the water.  While most people might be prepared to accept this sub-standard tea, a true connoisseur won't.  Then again, a connoiseur would probably not use mugs and teabags.  It also depends on the thermal properties of your mug - an insulated mug won't lose heat from the water, so there's no benefit to preheating it.  Similarly, a thin glass cup will lose heat so quickly there is probably no point heating it either.

The most notable example of tea not being brewed at the right temperature is, in my experience, tea from coffee houses.  Coffee is brewed at 92 degrees C, and they use this water to make the tea too.  8 degrees out makes the tea taste terrible.

I have studied this matter extensively and my own empirical data would suggest that warming the mug both improves flavor and speeds up brewing times and I would therefore recommend that you continue the practice. My data collection however was based upon my own subjective feelings as to what makes a good cup of tea and I'd therefore encourage you to conduct your own set of experiments.

The art of tea making is unfortunately dying out. I personally am no longer able to take part the collective rounds of tea making that go on in my office because of the nasty brew that my colleagues usually produce. This is unfortunate because as everyone knows the best cup of tea is one that someone else has made. The biggest contributor to their poor quality is undoubtedly the hot water urn which replaced the office kettle. The first rule of tea making is that freshly drawn, boiling water is required. Urn water is not boiling and often carries a metallic taint because it has been sitting in the urn for quite some time.

My final tip (and I think I picked this up from an interview with Kathy Burke that I read) is that you should always put 2 tea bags in the cup. This allows you to quickly make a good cup of tea without having to go through the palaver of waiting around while prodding and squeezing the bag to get the flavor out. This might seem like profligacy, but honestly life is just too damn short to spend standing around poking tea bags. In the final reckoning it is unlikely that anyone will wish that they have spent more time on such an activity. Also New Scientist kindly provides us with tea bags so I’m not paying for it!

We should of course turn to that most famour of tea snobs, George Orwell, who wrote an essay on how to make the perfect cup of tea, for a definitive answer. Or so one would think. I remember reading about a study by the Royal Society of Chemistry a few years ago that refuted some of Orwell's claims. Here are a couple of snippets I recall:

Add milk before the tea, not after (as Orwell recommended). Putting milk in afterwards destroys the milk's proteins and makes it taste bad.

Add sugar (a no-no for Orwell). Sugar makes it taste less bitter. Orwell liked the bitter taste.

As for warming the cup, Orwell always used a pot, to which he would add 6 spoons of tea leaves. Extravagant. But he made no mention of warming the cup. However, given his inclination for ceremony in tea making, I'm sure he would have readily accepted this as a worthy inclusion.

I do this too bacuae it means your tea is hotter and stays hot for longer. You just pour some boiling water in the cup and leave it for a minute. People say i'm werd because of it but at least now I know I'm not the only one.

As an avid tea drinker I shall give you a simple answer... Yes.

Why? Because you can, and it takes very little effort to swill a dash of hot water you've already boiled for the tea in your mug.

Now go make yourself a nice cup of tea and revel in the warmth the mug passes to your hands!