Are scripts like sp_BlitzIndex and the others accurate on restored backups? I am a developer trying to get a better grasp of things.  I am wondering if using the scripts from BrentOzar.com will work on restored backups?  For example if I have the DBA give a backup of production, then restore that on my dev box, then run the scripts, will I get accurate information or is vital information lost during that process?

I am guessing that the answer is both yes and no.  It looks like a good bit of the information that sp_BlitzIndex provides will work regardless since it seems to be based on the structure of the database.  I am not so sure about sp_BlitzCache, etc.  

I guess what I am asking is what types of things in these scripts won't work when running them on a restored backup and just as important, why not?

Thanks.
BlitzIndex will return some useful information on a non-production server as it is analyzing various things, some of which have nothing to do with what queries are running.  It should be clear to you when viewing the output what is relevant and what is not.

BlitzCache will not return useful information (or not much, at best) on a non-production server unless you run the same queries and load that are on the production server.

To answer your question as to "why not?", run them, review the output, and read the info at the URL returned in the URL column.  There is way too much info returned by these tools to give you a line-by-line description of why it would or wouldn't be relevant on a non-production server.