It probably won't be confirmed or relayed by many other nodes.
Confirmed? Sure, miners won't take that dust, if they have transactions with bigger fees. But relayed? Let's see:
$ bitcoin-cli getpeerinfo | grep minfeefilter
"minfeefilter": 0.00000000,
"minfeefilter": 0.00000000,
"minfeefilter": 0.00001000,
"minfeefilter": 0.00000000,
"minfeefilter": 0.00001000,
"minfeefilter": 0.00001000,
"minfeefilter": 0.00000000,
"minfeefilter": 0.00000000,
"minfeefilter": 0.00001000,
"minfeefilter": 0.00000000,
"minfeefilter": 0.00000000,
"minfeefilter": 0.00001000,
"minfeefilter": 0.00000000,
"minfeefilter": 0.00000000,
"minfeefilter": 0.00001000,
"minfeefilter": 0.00001000,
"minfeefilter": 0.00000000,
"minfeefilter": 0.00000000,
"minfeefilter": 0.00000000,
"minfeefilter": 0.00000000,
"minfeefilter": 0.00000000,
"minfeefilter": 0.00001000,
"minfeefilter": 0.00000000,
"minfeefilter": 0.00000000,
"minfeefilter": 0.00000000,
"minfeefilter": 0.00000000,
"minfeefilter": 0.00000000,
"minfeefilter": 0.00000000,
"minfeefilter": 0.00000000,
"minfeefilter": 0.00000000,
"minfeefilter": 0.00000000,
"minfeefilter": 0.00000000,
"minfeefilter": 0.00000000,
"minfeefilter": 0.00000000,
"minfeefilter": 0.00000000,
There are a lot of nodes, which declare to accept free transactions. Of course, a lot of them are probably not mining anything, but only relaying it.
However, if free, or low-fee transactions would be widely relayed by a lot of nodes, and they would be open for modifications, then guess what: users could use that kind of communication channel, to batch a lot of low-fee transactions into some bigger fee transactions, and then broadcast it to the miners.
But of course, allowing free transactions unconditionally means, that if all of these node operators are vulnerable to some DoS attacks. Unless of course they put some protections, and after accepting for example N free transactions, they stop accepting next ones, or they require a minimal Proof of Work inside some transaction, or something like that. But this is hard to detect, you can only see declared fees, it is hard to discover, if all of these nodes are really vulnerable, or if they just declare to be.
The average person would just be wasting time and end up with a stuck transaction that needs to be RBF-ed.
Note that RBF rules depends on fees. And guess what happens, if allowed fees are set to zero, and how RBF works in that case.
Not even all miners have agreed to include 0.1 sat/vByte transactions in their blocks, so I wouldn't go lower than that right now.
Why not, if there are a lot of nodes, accepting such transactions?