How to use the barrier method for equality (or positive) constraints?
My initial reasoning was that since the example there starts with constraints $f_i(x) \leq 0$ and then takes
$$\phi(x)=-\sum_{i=1}^m \log(-f_i(x))$$
Then if one'd start with $f_i(x)=0$, then reasonably one doesn't need to flip the sign so could one then simply do
$$\phi(x)=-\sum_{i=1}^m \log(f_i(x))$$
for constraints of the form $f_i(x)=0$ or $f_i(x) \geq 0$.
However, without being entirely familiar with the barrier method, is this enough?
A constraint $f(x) \geq 0$ is equivalent to $-f(x) \leq 0$. Of course $f$ should be concave if you expect globally optimal solutions.
An equality constraint does not fit in the barrier method, since the feasible set for such a constraint has a nonempty interior.
The page you link to also has a description of the primal-dual method. That method can cope with equality constraints (in fact, all inequality constraints are first reformulated as equality constraints).