No.
The republic was at best a third order issue in the minds of most people, probably on par with changing the flag i.e. a symbolic change that is nothing to get too miffed about.
Also, there is no constitutional way for a state to secede.
Economically it doesn't make much sense, with the lowest population growth and second highest unemployment rate amongst the states.
Finally, for every dollar that Tasmania contributes federally, they receive $1.58 in return in services and benefits. It's doubtful anyone would give that up.
State premiers and 'prominent' businessmen bring up secession in tough economic times. It costs them nothing, shows they have state pride and gets them a bit of good state press. Nothing ever comes of it.