Does putting a wireless router in a wooden box make the WiFi signal weaker?

This depends on the type of wood. Putting unconfirmed jokes aside, you can verify this yourself using a program like inSSIDer (linux version here) and compare the signal strength and signal to noise ratio's before and after putting it in the box.

There is an informative paper written in 2002 by Robert Wilson, a graduate student at the University of Southern California. The paper is called "Propagation Losses Through Common Building Material 2.4GHz vs 5GHz". In this paper he covers the transmissive and reflective losses of a number of materials, including plywood and particleboard. 18mm plywood has a transmissive loss of 1.9dB and a reflective loss of 9dB at 2.4GHz. At 5GHz the transmissive and reflective losses are 1.8dB and 30.5dB respectively.

His paper mentions the moisture of the wood affected the permittivity. Thinking about it, I would expect the explanation to be this: 2.4GHz is the frequency of a microwave oven. The radio antenna is most likely having the same effect on the moisture in the wood as a microwave oven has on food: The radio waves are being converted to heat by the moisture in the wood.

The paper can be found here: https://www.am1.us/wp-content/uploads/Documents/E10589_Propagation_Losses_2_and_5GHz.pdf

Ever since I changed my floor, from carpet to wood (maple 2cm thick), my wireless signal is WEAK (I live on 2nd floor, router is on 1st floor). I then changed my router (internal antenna) to another router (external antenna) but still have weak signal. I live in a house, so there isn't much interference from neighbors.

Weakened by 2 bars. Wifi analyzer about 10db. so YES it is possible it makes the signal weaker.