Has the Stanley Cup playoffs games ever been held on neutral ice?

Sort of. No complete series in the modern game was played entirely at a neutral site, but some series had one or two games played at neutral sites.

The 1925 Stanley Cup Finals, for example, was between the Victoria Cougars and the Montreal Canadiens. Back then, all of the games were held in one side of the country; alternating between West and East. That year, it was west, so hosted primarily from Victoria (which is in British Columbia basically between Vancouver and Seattle); but one game was in Vancouver, which was considered technically a neutral site (though not exactly neutral in any meaningful sense). Wikipedia calls it "the last neutral site game in Stanley Cup Finals history that did not involve the New York Rangers." Only one game (game 2) was hosted there, the other three in Victoria.

After that, the neutral site games I see were only when the Rangers were forced to play games on the road due to the Circus being at Madison Square Garden (this is akin to the Chicago (NBA) Bulls "Circus Road Trip" when United Center is occupied by the circus in the early fall). In 1950, in the NY-DET finals, five games were played in Detroit and two in Toronto.

This was the last Stanley Cup Final to feature a team that did not host any games and also the last to feature neutral site games, which were held in Toronto on account of scheduling conflicts at Madison Square Garden.

At least one other Rangers' finals games were played at neutral sites; in 1932, one game was played at Boston Garden in the 1932 finals.

Prior to 1925, it's likely there were other games at neutral sites, but you're getting far enough back it's not as relevant to today's game.