No, they do not get damaged at all, they may last longer because no current drawn.
One thing you could do is to add a lamp to reduce the current to float level
Measure the battery voltage disconnected from charging and loads.= Vbat
Measure the solar panel at full sun.= Vsolar
then Vsolar - Vbat = Vresistor
For each battery amp-hour capacity give 5 milliamps current.
So if you have a 50 amp-hour battery : 50 * 5 milliamps = 250 milliamps.
So Vresistor/0.25 = Resistor value that you place in series between the solar panel and the battery
Resistor power = Vresistor * 0.25 * 4 = Resistor wattage .
This way your battery will be fully charged when you return and the small charge you have will, as well, do a deep desulfation cleaning.
This procedure should be limited to about 3 weeks -- if you can measure the current you will read around 200 to 220 milliamps ( depends on the solar panel manufacturing process and the capability to supply low currents above the Vmp levels)
Nando