I understand that wheels should be trued for efficiency. Are there any safety risks (or risk of damaging the bike) in using untrued wheels?
Are the risks different between wheels untrued left-right and up-down?
My question is motivated by the following fact: I have an up-down untrued wheel and not much time to take it to the shop, should I keep using the bike?
It is important to understand that spokes work in tension rather than compression. What that means is that the spokes at the bottom of the wheel are not 'doing anything'. The load is shared by all of the spokes to varying degrees and, particularly with cross-laced spoking patterns, that load gets dynamically transferred to use the strength of the available spokes. You can actually knock out a chord of a dozen spokes on the cheapest of steel rimmed wheels and the wheel will still be able to support your weight and get you home, albeit with gingerly riding style with no kerbs or potholes attempted...
Therefore, even quite seriously mangled wheels that are as true as a poppadom are surprisingly strong when riding on paved roads in a straight line. As other contributors have noted the problem comes with the brakes and stability at speed. You really do want to make sure your brakes are not hitting the tyre or falling off the bottom of the rim. If you have the slightest doubt then you can ride the bike using the other brake as your primary, using the brake on the affected wheel for 'emergencies only'. This should not be a problem if the weather is dry and you know what you are doing.
Straightening out the wheel need not be something that you need to get the bike shop to do. Consider doing it yourself, with a spoke key on the wheel in situ. Here is a handy page on how to do just that:
http://www.kenkifer.com/bikepages/skills/spokes.htm
As for getting a spoke key, go into the bike shop with the bike and ask. There are a couple of common sizes and you will need to get the correct key for your bike.
Untrue wheels are weaker, may cause steering to be difficult at high speed, and if you're using rim brakes it will cause stuttering or lockups. Having said that, it would need to be damn untrue for any of the above to be a serious problem.
It depends on the reason for them not being true. A lack of equal tension in the spokes could mean weakness in one (or more) of them - and broken spokes are not a good thing to ignore. You can survive one or maybe two for a short while, but eventually the rim could be in danger of collapsing.
If you have rim brakes then left-right could mean mismatched application of the brake blocks which could make for inconsistent pressure but also it could attempt to move the wheel from side to side, which might start to work it loose at the hub - and loose hub nuts are very dangerous. In this situation up-down is also sub-optimal as the brake blocks could be rubbing on different parts of the rim (or even the tyre).
I was going down some serious hills on a road bike with warped rims. At some certain speed (no speedometer) there was a considerable amount of wobble that started to happen. An unbalanced mass rotating at high speed can cause a significant amount of wobble. This is how the vibrate feature in your phone and game controllers work, by attaching an unbalanced weight to a small electric motor. Let's just say this is something that I would not want to experience again. Took it slow the rest of the ride.
Mostly, the out-of-true up/down wheel simply makes your ride less pleasant. Out of true side-to-side will cause the brakes to drag, if severe enough, and make them grab a bit on light/moderate braking, even if they aren't dragging normally. And either will make the bike harder to handle at high speeds (especially those 50mph downhills).
In theory, an out-of-true wheel probably affects spoke life, and a wheel, once out-of-true, will tend to get worse over time. And, of course, if a wheel goes suddenly out of true it may indicate that a spoke is near failure.
But none of this (other than the handling problems, which you can evaluate yourself) represents a real hazard, and there is no great harm in continuing to ride on the wheel if it's not too unpleasant to do so. It's not like a loose crank arm or some such where failure to fix it immediately can cause serious additional damage.
All that said, a decent quality wheel in good condition, once trued for the 2nd or 3rd time (ie, fully "broken in") should stay true for years, if you don't subject it to any severe shocks. So it's worth paying to have it done (or learning how to do it yourself) to help assure years of pleasant riding.
To add to the sage advice aleady given, an untrue wheel at high speed will wobble causing increased risk of pinch flats, especially near the valve. I know because it had happened to me just recently.
