Why do they use paper to print dollar bills instead of coins?

The US continues to print $1 bills mostly due to psychological and political reasons.

First, unlike other countries, the US has rarely made significant changes to its money during the last century, except for eliminating gold coins in 1933. Many Americans see changes to coins and bills as representing instability rather than improvements or novelty, and the withdrawal of the $1 bill from circulation would be interpreted by many as an admission that the dollar's purchasing power has declined. The dollar has lost 90% of its purchasing power due to inflation regardless of whether it's made of paper, metal, or Saran Wrap, but that reality hasn't been matched by perception because the bills still look the same.

Second, the $1 bill has used essentially the same design since 1928 so it's achieved a kind of icon status, a bit like Mt. Rushmore or the curly script on a Coca-Cola bottle. Again, no one likes to change an icon.

Third, early efforts to reintroduce the $1 coin were clumsy and flawed. The Eisenhower dollar was too big and the SBA dollar had too many things wrong with it to list here. Those two failures set the stage for the rejection of ANY $1 coin, regardless of how well designed it might be or how successful other countries' $1 coins have been.

Finally, the company that prints all paper for US currency has developed an extremely cosy relationship with the government. So many $1 bills are printed that nearly half of their business comes from that one denomination. Whenever there has been a proposal to stop making $1 bills the company immediately lobbies Congress and the politicians go to bat with protective legislation.