Henadic Theologian
Legend
No, not true. It is absolutely stated in pre-Avatar Trilogy materials that the gods need worship (1981's "Down-to-Earth Divinity" and the 1e FRCS, for example, both written long before Ao was invented). Similarly, later works such as Faiths & Avatars establish that gods would decline and die from lack of worship before the Time of Troubles, most famously in the case of Amaunator.
It is true that in the novel Waterdeep (the third book of the Avatar Trilogy), the speech Ao gives includes the declaration that he's imposing the need for worship as a new punishment for the gods. This is a continuity screw-up by the author and editors of that book, and should be ignored given the consistent evidence, both pre- and post-Avatar publication, that the worship reliance existed pre-ToT.
I was aware of the continuity issues at least partially, but its all trumped by the fact that Gods like Selune and Shar and others predate their worshippers by possiblity thousands or more years, they predate Toril.