No. Truecaller never reads or monitors the content of your SMS messages. Truecaller only analyzes the message locally on your phone to determine if it is spam or not, and to identify the sender for you. This is all done on your iPhone without touching Truecaller servers. Even if Apple allows more access, Truecaller only needs the number, and no information leaves your phone.
In case you wish to change the settings, you can simply visit: iPhone Settings > Messages > Message Filtering > Unknown & Spam.