Most of the research that I can find is inconclusive, and deals with networks that run on different specifications than home WiFi.
Basic Technology Background
There is some research on ELF (extremely low frequency, like from power lines or appliances). This is generally below 300 MHz (and much lower)
Another area that I can find research on are cell phone networks. These run generally from 900-1800 MHz. A cell phone tower generally has an effective radiated power of 5-10 W (right in front of the antenna, it falls away pretty quickly). A cell phone transmits between 0.5 W - 3 W (sources vary, depends on manufacturer, etc).
For comparision, WiFi is generally run at 2.4 GHz (2400 MHz) or 5.6 GHz (5600 MHz). A WiFi basestation's specifications say that it can have at most 20 dBm (0.1 Watts) on each of the frequencies that it runs.
Balmori and Hallberg are two researchers who studied sparrows in Spain in the mid 00's. They determined that sparrows were in generally poorer condition and had worse breeding outcomes when electronic magnetic environment increased.
Everaert and Bauwens did a similar study in Belgium (with similar results). The problem with these studies is that it is difficult to determine how much of the decrease in health/outcomes is due to the EMF and how much is due to increasing urbanization (loss of habitat, food sources, etc).
Reijt et al. (2007) studied breeding tits in nesting boxes around a military radar station compared with a control location. The exposure levels were reported as being from 2.0 to 5.0 W/m2. No change in breeding biology was observed. However there was a shift in the ratio of blue tits to great tits compared with the control location. Thus one interpretation of this study and those on house sparrows is that the electromagnetic fields may discourage some bird species from breeding there or alternatively might encourage other species to build their nests in the areas with higher RF EMF fields.
If you want to look into this more, I suggest including the term "non-thermal" as a search term. Most of the current safety guidelines discuss the ability of the energy waves to heat flesh. There are no concerns with the thermal safety of any of these forms of radiation, but non-thermal effects are still being studied.