"You get what you pay for."
This can largely encompass the mindset of non-paying clientele. They may "settle" where they normally would not, because they aren't paying for the work. So, there's a consideration there when working for free.
As @ZachSaucier pointed out effectively.... there's also the lack of timelines, restraints, and traditional demands a paying client brings.
You can spend months, in some cases years, working on something for free because the client isn't concerned with anything other than getting what they want. Invoices prevent that. Invoices force the client to be aware of time and demands. And it's how effective a designer is under demand that is valuable to review. Even the most inept designer will eventually stumble upon something great.
I also agree with Zach in that "non-profit" does NOT ever mean "no money". Non-profit or not-for-profit CEOs make HEFTY salaries. These organizations DO have money. Their tax structure and budget is not your concern and never should be. I, personally, always look at non-profit as merely a tax structure. What importance is it to me, the designer, if the organization pays less taxes??? That's all "non-profit" really means.
That being posted you can invoice for time as opposed to dollars. When I've volunteered my services it has been under a time restraint and I've donated X hours of my skills. I then give the client time estimates rather than dollar estimates. The client can then choose to use my time in whatever manner they feel will benefit them most. And I invoice for "hours spent" after any project. This lets the non-paying client know.. they have X hours left.
This method has been effective for me in limiting the back and forth, constant revisions, and otherwise lack of consideration most non-paying clients inherently bring. One must be diligent is sticking to the hours donated though. It's all too easy to "throw in" a few more hours to complete a project. The designer should think of this in terms of money (i.e. billable hours).... you wouldn't return to a client and invoice for $XXX more because you "threw in more time" to complete their project. The client would be unhappy. So if you donate 10 hours, you ONLY work 10 hours.
As for how the work is seen...
Yes if you show open-ended, non-paying, get-it-done-whenever, work it's not as favorable as showing work in a portfolio which was bound by restraints. Again, how a designer performs under demand is more telling overall. And it's any lack of demand that tends to devalue a portfolio piece overall.
Most employers are seeking someone to complete projects in a timely manner with some given client demands. Explaining that you donated X hours to OrganizationY and this is how you spent those hours meeting their demands for X, Y, and Z does give insight into how a designer will perform under time restraints. And X hours of work can be translated to dollar amounts for any potential employer.
This provides a method of showing free work which is bound by much of the sane restraint paying work would be bound by.
You will come across clients, paying or otherwise, that will abuse you to some degree. Expecting more time from you or more effort than would be traditionally be necessary. You should be willing to walk away from these clients. More so if they are non-paying clients.