It seems possible ...
WARNING! - Refurbished only means you will get a working phone back.
Even if they guarantee to return it with your data intact, be VERY CAREFUL! they are just telephone call handlers and do not know for sure what will happen, also people do not tick all the correct boxes and others do not look at all the boxes, so you could loose out those ways!
I am very surprised that there is no information on this subject.
It must have been an issue for 1000's of people over the past few years.
My problem is that "back up has stopped", would not re-install, and I could not find out why (it said memory, but there was loads free - since found it was talking about Dalvik).
Over the weeks I built up a collection of useful information and photos which I do not want to loose.
I am assuming yours is an Android phone, but even if it's a Windows phone, the principle is still probably the same.
It appears to be possible to recover files from the internal memory of a dead phone using a PC, free software and USB cable.
The principle is start the Flashing process, BUT do not Flash the phone, because I can not be sure it would not destroy your data / files.
When at the stage to be able to Flash the phone, effectively your PC thinks it is communicating have a "USB memory device".
This is achieved by removing the phones USB driver and substituting another driver (to enable the Flashing process).
I've not had the time to spend on it that I need.
I will come back when I do finish ...
BUT do not wait for me, there seems to be too many things going wrong, and the most important get done first. Which is why I have things waiting for over 1 year now.
One problem I have is that Windows keeps throwing out the "substitution USB driver" and replacing it with the original phone USB driver.
Windows thinks the phones driver is more suitable, so automatically re-installs it.
I know the issue, I've had it several times previously with the "unknown device USB driver" (a crash causes the driver change). I just need to find the "Windows stored back up driver", delete it, then Windows will happily accept the "substitution driver".
Ahhh, the joys of owning a MS Windows computer :-(
Some usefull software names :-
USB VCOM Driver
Windows drivers - A25_USB_VCOM_Drivers
usbdeview_droidtricks
Some useful search terms :-
brick
unbrick
flash rom
SP Flash Tool exe v3 (not to use, but the process' before is what's wanted).
Searches with search engines (Google, Yahoo, etc.), also You Tube (I could only find Flashing, but as previously, it's the process up to the "Flashing point" that you are interested in).
Sorry it's unfinished information, but at least it gives you hope and some where to start.
Data Recovery firms will be able to recovery the data / files and some are open to "please help, I'm skint ...", by reducing their prices to a poor private person, but they will not do it for free.
So good luck to all of us.