I know the temperature, $T$, in $\mathrm{eV}$ and density, $n$, for the three ion components of a chosen plasma, say $\mathrm{O^+}$, $\mathrm{He^{++}}$, and $\mathrm{H^+}$. But I would like to know the temperature for the total ion component, $T_\mathrm{tot}$. Would it be wrong to use:
$$T_\mathrm{tot} = \frac{\sum_i{T_i \cdot n_i}}{\sum_i{n_i}}$$
where $i$ is the $i$-th ion component?
Your expression for the total ion temperature works for a mixture of ideal gases. That is, if one assumes the thermal pressure of species $s$ is given by: $$ P_{s} = n_{s} \ k_{B} \ T_{s} \tag{1} $$ where $n_{s}$ is the number density of species $s$, $k_{B}$ is Boltzmann constant, and $T_{s}$ is the temperature of species $s$. Then the total pressure is given by: $$ P_{tot} = \sum_{s} \ P_{s} = n_{tot} \ k_{B} \ T_{tot} \tag{2} $$
Thus, if the pressures follow these approximations, then yes you can define the total ion temperature as: $$ T_{tot} = \frac{\sum_{s} \ n_{s} \ k_{B} \ T_{s}}{n_{tot} \ k_{B}} \tag{3} $$
This all breaks down, of course, if the ions cannot be approximated as ideal gases. However, it is not necessarily the case that the partial pressure approximation is completely inappropriate.