Hello student student as per the NTA specification yes Date is mandatory on neet application photo.

Specifications by NTA are mentioned below -

Date on photo should not be later than September 1 ,2019

Ears should be clearly visible

80 % focus on face

The photo must be in white background

There should be nothing in the background

Caps and goggles are prohibited

For more updates refer to the link mentioned below

https://medicine.careers360.com/articles/neet-2020-photo-size-format-documents

Feel free to comment if you have any doubt

Good luck