I have two main comments
The first is that you do not need to nest in QuickBase. It has esier syntax than Excel
Just list your conditions vertically (for readability) and Quick Base will evaluate them in sequence and take the result of the first one that is true.
The next issue is that you this is a text formula, so you need to rethink where you have a result which is a checkbox boolean true result. That does not make sense as a text result and will give you a syntax error.
If(
[Permits]="Yes","Permit Pending",
[Reassigned to Drops] = true, "say something here",[New Build] = true, "say something here",
not IsNull([Date MDU Splicing Completed]),"Splicing Complete",
[Access Issues]="Not Resolvable","ROE Impasse",
not IsNull([Date MDU Temp]),"Temp",
not IsNull([Date MDU Ready to Splice]),"Ready to Splice",
not IsNull([Date MDU In Construction]),"In Construction",
not IsNull([Walkout/Survey Date]),"Walkout/Survey","Other")
Be sure to consider the sequence as the first true result will be the result.