What happens to a metal plate in the photoelectric effect? In the photoelectric effect the electrons are supposed to be removed from the plate if light of appropriate wavelength hits the plate. 
If electrons are removed, the plate should get ionized (positively). If this is true, does the plate change its properties?
 A: You're quite correct that the plate will get a positive charge, but then any piece of metal can be positively or negatively charged in many ways. For example a Van de Graaff generator does exactly this, as does a gold leaf electroscope. The charges involved are tiny compared to the number of mobile electrons in the metal and make no significant difference to the properties of the metal.
In the case of photoelectron emission from a metal surface, the electrons leave the metal with a maximum energy of:
$$ E_{max} = h\nu - \phi \tag{1} $$
where $\nu$ is the light frequency and $\phi$ is the work function. However as the metal becomes charged it develops a potential $V$ given by:
$$ V = \frac{Q}{C} \tag{2} $$
where $Q$ is the charge on the metal and $C$ is its capacitance. The electron requires energy $eV$ to escape from this potential, where $e$ is the electron charge, so the maximum potential you can create by photoelectron emission is:
$$ V_{max} = \frac{h\nu - \phi}{e} \tag{3} $$
At voltages higher than this the photoelectrons don't have enough energy to escape so you can't increase the charge on the metal.
Since the charge is just the number of photoelectrons times the electron charge you can combine equations (2) and (3) to tell you the maximum number of photoelectrons that can be ejected:
$$ \frac{ne}{C} = \frac{h\nu - \phi}{e} $$
so:
$$ n = \frac{h\nu - \phi}{e^2}C $$
A: You would be right that the plate would get positively charged (if this experiment was conducted in vacuum). However, if this was the whole story, that would make it more and more difficult to emit the next electron and even harder yet to emit the next one and so forth, as the plate would get even more positively charged. 
Therefore, to avoid this scenario, most times the photoelectric experiment is done, the metal plate is electrically attached to a replenishing source like earth ground or a battery. This keeps the metal plate neutral at all times and keeps the potential barrier the same for all the electrons.
