No.

This is a weird persistent "fact" that people love to bring up about the American Revolution and I've never been able to find a primary source that supports it. The earliest mention of the idea comes from an article in Harper's Magazine in 1896 -- a writer named Brander Matthews wrote about how odd it is that Franklin DIDN'T write the Declaration, and decided that it must have been because his colleagues were afraid he'd slip a joke in. (That article is called "The Penalty Of Humor" and is in Harper's Magazine Vol. 92.) It's a good story, but the truth is more complicated.

There was a committee of five chosen by Congress to draft the Declaration -- Thomas Jefferson, John Adams, Benjamin Franklin, Roger Sherman, and Robert Livingston. The actual committee never took minutes of their meeting, and there are discrepancies between the accounts of Adams and Jefferson because the Declaration ended up being a much more influential document than anyone expected. At first it was viewed as an almost tedious administrative task; they weren't drafting the most important founding document in American history. In Founding Brothers, Joe Ellis suggests that Adams, had he known how important the Declaration would become, would have written it himself. (Adams's own writings support this, in my opinion -- later in life he insists that he rallied a bunch of others to vote that Jefferson write the Declaration, thus taking some of the credit for the document. "I chose Jefferson to write it because I knew he'd do an awesome job" sounds better than "I made Jefferson do it because I felt it was beneath me." But Jefferson's recollection is that it was foisted on him because Adams couldn't really be bothered.) Ellis also says that Franklin loathed having his work edited by others, which is another reason he probably didn't care to write it. But the entire committee reviewed and made edits to Jefferson's original document, so Franklin was definitely involved. But no, he wasn't prevented from writing the whole thing because everyone was afraid he wouldn't take it seriously. That's a myth of Parson Weems proportions.

(I've never commented on a thread in AskHistorians -- mods, I apologize if my sourcing isn't sufficient. I don't have my copy of the book on hand. Please delete if necessary!)

Edit: here's a link with an excerpt of Adams telling the "I insisted Jefferson write it!" story.

Hey guys, comments are likely getting deleted because they contain conjecture and lack of sources.

If you're going to comment, please have info to back it up--saves everyone a lot of time and keeps this subreddit's info standard high.

As t he discussion wandered into important questions about the authorship, contemporary meaning and legal significance of the Declaration, I suggest reading Pauline Maier's celebrated and authoritative work, American Scripture: Making the Declaration of Independence (1997), and the brief summary she provided for a paper edition of the Declaration and Constitution. The Declaration was important as a political fact; the concluding paragraph that contains the actual declaration was what seemed important, and was read at Independence Day celebrations for decades. The now-famous opening paragraphs became famous and important later. Maier takes Jefferson down a bit for exaggerating both his role and the importance of his contributions to those opening paragraphs. . . . . As to legal significance, lawyers today do not consider it to be a legal document, but there is an argument to be made that it was constitutive, possibly therefore part of the constitutional canon. Most of the state constitutions, including those that preceded the Declaration of Independence, began with declarations or rights, and Jefferson drew on those, especially the Virginia declaration of rights drafted primarily by George Mason.