You can control whether added graphic element to a plot are drawn before or after the plot is generated by choosing between
Prolog and
Epilog options.
Compare
Graphics[{Red, Disk[]}, Prolog -> {Green, Rectangle[]}]
with
Graphics[{Red, Disk[]}, Epilog -> {Green, Rectangle[]}]
However, drawing white lines over parts of the axes to cover portions of the underlying aready-rendered axes may have unintended consequences (AKA, it's a hack). A better way for you to do this is to set the option
AxesStyle -> Directive[Opacity[0]] and then add the portions of the axes perhaps along this approach:
Plot[x, {x, 0, 3}, TicksStyle -> Directive[Opacity[1]],
AxesStyle -> Directive[Opacity[0]],
Epilog -> {Line[{{{0, 0}, {1, 0}}, {{2, 0}, {3, 0}}}],
Line[{{{0, 1}, {0, 2}}}]}]
