The wicketkeeper-batsman tore his left hamstring, the country's cricket board (ECB) said on Wednesday.
Foakes (28) was set to play in what would have been his first home series before sustaining the injury when he slipped in the dressing room after Surrey's County Championship game against Middlesex.
He is expected to be out for at least three months.
James Bracey is now due to make his debut behind the stumps at Lord's, with England coach Chris Silverwood calling up Kent's uncapped Sam Billings as Foakes's replacement.
Nottinghamshire's top-order batsman Haseeb Hameed has also received a call-up.
England's regular keepers Jos Buttler and Jonny Bairstow are both being rested following their involvement in the Indian Premier League.
The first New Zealand test starts on June 2 (local time) and the second at Edgbaston on June 10.
Foakes' injury may also rule him out of the start of the five-test series against India that begins on August 4.
