World number 32 Boulter entered her quarter-final in fine form and knowing she will be seeded for the first time at Wimbledon, which begins on Monday.
But she endured persistent issues with her ball toss and a couple of double faults in the fourth game saw her serve broken before a further double fault in the wind led to another reverse as she lost the opening set 6-1.
French Open runner-up Paolini broke again in the fifth game of the second set before Boulter drew level at 4-4.
The British number one forced a tie-break but was beaten to love as her semi-final hopes were dashed.
Dart faced Fernandez, who was runner-up to Raducanu at the US Open in 2021.
Although unseeded, the Canadian is ranked 75 places higher than Dart at number 30, and made her class tell in a quickfire victory despite losing her first service game.
Fernandez will face defending champion Madison Keys in the last four after 2023 French Open finalist Karolina Muchova withdrew because of a right wrist injury.
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RankDriverNumberTeamFastest LapLaps11Lando NorrisNOR44McLaren1:33.9031:33.9036622Charles LeclercLEC1616Ferrari1:34.0391:34.0397733Lance StrollSTR1818Aston Marti