No, however, you will not get notifications via smartphone if it's not connected the the Nest app. The purpose for a "smart" fire/CO device is to alert you when not at home via the app. In addition, it has other feature like automatically checking the devices batteries and sensors. The device will always alert you locally with an audible alarm for fire or CO, even without WiFi/internet.
Here is an excellent article: https://www.howtogeek.com/362600/will-the-nest-protect-still-work-without-a-wi-fi-connection/
Here is the Nest Protect setup guide: https://nest.com/support/article/How-do-I-connect-my-Nest-Protect-Wi-fi-and-my-Nest-Account