Temperature effects on the electrostatic instability of non-symmetric electron plasma systems consisting of two warm counter-streaming beams and of warm background particles are investigated linearly (analytically and numerically) and nonlinearly (by computer simulation experiments) for the case of Heaviside and moderately warm Maxwellian particle distribution functions. The non-symmetry is due to unequal temperatures, streaming velocities and particle densities in the beams. Other variable parameters investigated are the relative thermal velocities of the beams and background as well as the relative background particle concentration. When the beam temperatures are unequal, unstable waves with Re ω > 0 and propagating in the direction of the beam with lower temperature occur; this is in contrast to the equal temperature symmetric two-stream case, in which the unstable waves have Re ω = 0 (standing waves) and the temperature effect is only to decrease the growth rate. When three warm components are present in the system, the following results hold: (i) the beam temperatures have the effect of decreasing the importance of the unstable standing waves with Re ω > 0 (and growth rate γB) relative to the waves with Re ω = 0 (growth rate γA) which occur in cold three-component symmetric systems; in addition to this, both γA>max and yBpTaax decrease with increasing temperature; (ii) the background temperature has the general effect of reducing the absolute maximum growth rate. For relative background temperatures above a certain critical value, a separation (in k and ω spaces) of regions B and A occurs; γAmax increases and γB,max decreases with increasing relative background temperature.