First of all, the pure (external) exposure of wood to nuclear radiation causes no problem as no radioactivity is generated in the wood by this radiation. However, wood from high contaminated areas will have a certain uptake of radionuclides, especially  Cs-137, which will still be present today.

There are estimations in IAEA Tecdoc1376 about different scenarios using wood from the most contaminated areas in the Chernobyl region (see link). Some of these scenarios can cause exposures above radiological permissible limits (table XVII). However, these scenarios  comprise mosty the use of wood ash for ferilisation and/or an extensive use of wood for building etc.

For all practical reasons, a single piece of wood from the Chernobyl region won't cause harmful doses, even at the highest radioactivity levels for wood given in table XVI. A piece of wood, let's say, one kg with an activity of 10000 Bq/kg  at a distance of 10 cm will roughly cause a dose of 0.1 mSv per year.  However, depending on its exact activity, you might be obliged to handle it within a controlled area. 

http://www-pub.iaea.org/MTCD/publications/PDF/te_1376_web.pdf

I agree with Freddie. exposure of wood by radiation has no effect but we should be careful in handling of contaminated wood

The exposure of wood to neutrons is not likely to produce significant induced radioactivity.

There are no danger if the wood exposure to Gamma, Alpha ,Beta radiations for neutron we should be careful.  but if there are any contmination on the wood then we must take care 

It depends the types of the nuclear radiation. If the radiation source is gamma rays, it does no matter to the wood. However, if the source is neutron, the elements of the wood may be  activated. It means that the wood becomes a radiation source, so you shoud tackle it with some safety protections. Furthermore, if radioactive materials have been dropped on the wood, you should deal with it as the case in neutron contamination. 

yes it is safe

Wood exposed to nuclear radiation won't activate the wood, however if its in contaminated area then there are cretin detectors to check the activity!! and with handling you can use special gloves...

You should remember that  your presence in nuclear contaminated area is very danger, So TAKE CARE!!