If you do not specify a Team and Provisioning Profile when building iOS apps on Xcode, you will get an error.

In this case, can't I run or build without a Developer account?

In conclusion, that's right.

Developing apps using Apple's Push Notification Service (APNs) requires steps such as registering the App ID of the app you are developing and creating a certificate for that application ID, otherwise the provisioning profile required for the build will not be automatically generated.

In order to perform the above steps, you need a registered Developer account with the Apple Developer Program, so you cannot build without it, or of course you cannot.

Local/Remote Notification Programming Guide
　Provisioning and Development

アプリケーションの配布に関するガイド
　　プッシュ通知の設定をする

If you search around iOS Push Notification Development Procedures, you will find an article in Japanese that provides detailed instructions, which may be easier to understand than Apple's official document above.

Unfortunately, for apps with push notification, there is no way to create a trial project without a Developer account and check the operation.