What effect does the resistance of the fuse have on the output voltage of a voltage divider?
Theoretically the output voltage should stay the same, but will it
  change at all in a real world scenario?


Yes it will, because any real world fuse has resistance. And this resistance will also change with temperature.

Next question: how much resistance has a fuse? Well, it depends on its type and on its voltage, current and \$I^2t\$ ratings. The nominal "cold" resistance (i.e., at &lt; 10% rated current) can range from &lt; 10 milliOhms up to several Ohms.

As an example, look at this Littelfuse catalog. A sample page from it:



Conclusion

you must select an appropriate fuse for your application in order to avoid having any significant and undesired voltage drop in it when connected to your load.