If you have a somewhat clear idea about what you consider to be 0 effect, you can compute the probability that the posterior variable is above that threshold (e.g. 1e-3) and take that as the probability of the effect not being zero.
In general I’d recommend plotting the ecdf so it is easier to see how much mass is above 1e-3, and above 1e-2, and above 0.2… you can use az.plot_ecdf for this. The only trick is that for now (and for 1-2 more weeks until the next arviz release) you need to install the arviz development version