The Russian adverbial participles have relative tense, that is, the present tense adverbial participles denote a secondary action which is simultaneous with the main action of the sentence, and the past tense adverbial participles denote an action which was / is /will be prior to the main action.
In this respect your first example is OK, "I got tired while reading a book". The second example sounds strange, though. If you need to say "I got tired after reading a book" or "Having read a book, I got tired", then you should better use the past adverbial perfective participle прочитав and you will have a nice sentence. But if you do want to use the past adverbial imperfective participle читав, then the sentence will mean the same as your first example, but it will sound too archaic, kind of 18th-century-style. In the Modern Russian, the past adverbial imperfective participles are used mainly in the negative form, with the particle
не, their positive forms are considered obsolete and are not used any more. Here are a couple of examples of Modern Russian sentences with the negative form:
И как многое пропадет для того, кто пришел слушать симфоническую поэму Рихарда Штрауса «Дон Кихот», никогда не читав романа Сервантеса... [И. Л. Андроников. Разные грани // «Юность», 1970] — "And how much will be lost for the one who came to listen to Richard Strauss' symphonic poem 'Don Quixote' without having ever read the novel by Cervantes..."
Она рвала его письма, не читав их. — She tore his letters without having read them.