Anti-static bags - safe to support powered circuit boards?
There are different bags available. Some of them are  insulators (nearly, see below) and only prevent the build-up of static charge. Others are conducting (grey-metallic ones typically or black ones).

I wouldn't place any powered electronics on conducting anti-static bags or mats. Even the fairly high resistance can have a big influence on the electronics (depending on function and design). 

Low powered stuff often works with fairly high resistance values, so the effect of parallel resistance there is quite profound.

I often use a sheet of paper to put my circuit boards on (note it can be charged, so make sure to wipe the charges off). Or just a holder where you clamp it from both sides so the main area has no contact at all (like when it is installed somewhere).



Update:

I'm not an expert on ESD and anti-static materials, let me try anyway:

Materials are divided into different classes depending on their sheet resistance (Ohms per square):


\$10^{13}\$ and greater: insulators and basic plastics
\$10^{9}\$-\$10^{12}\$: anti-static
\$10^{5}\$-\$10^{9}\$: dissipative
\$10^{3}\$-\$10^{5}\$: conductive


Everything below that doesn't belong in the field of ESD anymore. So the insulators and basic plastics are not able in any way to let charges move on their surface or through them. If they get charged, the charge stays there until it is dissipated in some way (discharge on you or an IC, in humid air it will decrease over time).

Anti-static materials have still a very high resistivity, so that you probably are not able to measure it with a normal multimeter. They are just a little bit conducting, so that the surface charge cannot build up. The triboelectric effect (charging through rubbing something etc.) is prevented. This gives a very basic protection. You can easily zap through a bag with a static discharge. These bags are usually pink in color, they are made of a pink polyethylene. 

This is what I had in mind when I wrote "Some of them are insulators and only prevent the build-up of static charge." So it's not true. It's just that the resistance is very very high (gigaohm range) and unlikely to cause a problem, but if the resistivity is at the lower end of the range, it might. 

I recommend watching the two videos from EEVblog: #247 and #250. Dave actually shows that the mats are not conductive enough to make problems with measurements.

One thing which is mentioned in the research is that the materials used do not behave as a simple Ohmic element, so the resistivity can be voltage dependent. The stuff gets usually tested at 500V to take that into account, but you never know.