If there's no change in temperature how can we say the heat supplied
to or the heat released from the system is non zero?
Heat and temperature are not the same thing. Heat is a mode of energy transfer whereas the latter is a measure of average kinetic energy. Heat flow may or may not change the temperature. If the heat add in to the internal kinetic energy of the system you will observe a change in temperature but if something else happens while, heat is being transferred, which robs the system of its internal kinetic energy, you will observe no change in temperature.
For instance, consider a closed system with moveable piston. If there is no net change in temperature and you wish to find about heat transfer to (or from) the system, you have to look at the work done by (or on) system. According to the first law of thermodynamics,
$$\Delta U=q-P\Delta V$$
For an isothermal process,
We have a given condition that the temperature of the system is constant. So if something happens in the system which increases its $U$, something else must happen which could nullify this increment, so that the net effect of such a process brings no change in the internal energy of the system.
Heat supplied or work done on the system increases the internal energy and heat released or work done by the system decreases the internal energy. So if heat is supplied to the system, work has to be done $by$ the system, if the temperature has to remain constant. And these two energy terms operate opposite to each other.
State change (like the melting of ice )at constant temperature, heat
is supplied to the system and the system is in thermal equilibrium,
how is work being done here?
$\Delta U$ is zero at constant temperature only for ideal gases. Internal energy contains both kinetic energy and potential energy terms, but since ideal gases do not interact with each other, the potential energy term is removed.
However, in the real world potential energy comes into play. The heat supplied externally adds in to the internal potential energy of ice. No change occurs in the internal kinetic energy of the system, which implies that the temperature remains constant $(\Delta T=0)$.