With your choices for gridrefine, crossingrefine, numpoints, rational, etc, you can still suppress the line through the additional signchange=false option.
The signchange=false option instructs implicitplot to exclude points on the line x=-mu across which F1 changes sign without attaining zero.
It's not clear whether you bumped all those options as attempt to fix your issue, or for some other reason. Perhaps you actually have some other example, since this one can be solved explicitly and plotted much more efficiently with the plot command and its discont option.
I decreased the very extreme gridrefine and crossingrefine from 15 to 3, which would still exhibit the line without signchange=false. But at 15 a great deal of work is done for practically no benefit.
F1:=1/(x + mu) + x - 1:
plots:-implicitplot(F1 = 0, mu = -8 .. 8, x = -4 .. 4,
color = blue, thickness = 2, rangeasview,
gridrefine = 3, crossingrefine = 3, rational,
numpoints = 100000, resolution = 3000);
plot([solve(F1,mu),x, x=-4..4], discont, view=[-8..8,-4..4],
color=blue, thickness = 2, labels=[mu,x]);