The 20-year-old has struggled to shake off the persistent problem and will now be replaced by Jocelyn Rae, who is two years her senior.
GB captain Judy Murray said: "While we are obviously going to miss Laura on and off the court in Budapest, it's a great opportunity for me to bring Joss into the team as a doubles specialist.
"She has shown her potential winning gold at the Delhi Commonwealth Games in 2010 and is back to full fitness."
The Fed Cup takes place from 4-9 February.
- Sports & Recreation
- Heather Watson
- Johanna Konta
- Jocelyn Rae
- Judy Murray
- Tara Moore