Various reasons are pertinent to the websites which do not allow creating an account with a temp mail address. These are:
Spam: Temp mails are usually issued to launch numerous accounts which will virtually fill a site with a fictitious or inferior sign-up. Companies have to necessarily implement these in order to cut down any amount of spam and fraudulent activity from invading their place.
Security: Temporary email addresses are used for creating phishing accounts, and frauds and even bots. Removing these helps companies in protecting their users and maintaining a better experience for everyone.
Account Integrity: Actual emails are favored as many users who are more likely to use the service longer offer personal and payment information. A temporary email does not offer any means of contacting or verifying the user after an occurrence where the account experiences troubles.
Abuse and Fraud Prevention: It can also be used to create new promotional offers which are unfair or migration which is a form of abuse. These serve as an advantage as the companies will be trying to mitigate these exploits types.
Service Quality: There are some services that mandate a working email address so that it can be used to reach the customer for account recovery and others. There is a high chance that the temporary emails used will not be of any help to the users and the negative experience will result.
In the event that using a temporary email is impossible, this is likely due to the fact that the site is attempting to protect the quality and integrity of its user community. You would have to provide some personal legitimate email for verification purposes.