There is no proof that this is the case
While I found extensive citations that this legend exists (e.g. Wikipedia calls it a myth), it is clearly not possible to prove a negative.
The Wikipedia article cites a book which I cannot check myself: Koch, Ebba. (2006) The Complete Taj Mahal: and the Riverfront Gardens of Agra. Thames & Hudson
The official web site of the Taj Mahal has a fairly complete historical section and a legends section.
In the historical section, it mentions the names of the actual workers without mentioning that their hands were supposedly cut off.
A labour force of about twenty thousand workers was recruited from across the Northern India. Sculptors from Bukhara, calligraphers from Syria and Persia, inlayers from southern India, stone cutters from Baluchistan, a specialist in building turrets, another who carved only marble flowers were part of the thirty-seven men who formed the creative unit. Some of the builders involved in construction of Taj Mahal under the master supervision of the Emperor Shah Jahan himself are:
- Ismail Afandi (a.ka. Ismail Khan) of the Ottoman Empire — Designer of the main dome.
- Ustad Isa and Isa Muhammad Effendi of Persia — Credited with a key role in the architectural design.
- 'Puru' from Benarus, Persia — Mentioned as a Supervising Architect.
- Qazim Khan, a native of Lahore — Cast the solid Gold Finial.
- Chiranjilal, a lapidary from Delhi — The Chief Sculptor and Mosaicist
- Amanat Khan from Shiraz, Iran — The Chief Calligrapher
Besides the above, Muhammad Hanif, a supervisor of masons and Mir Abdul Karim with Mukkarimat Khan of Shiraz handled finances and management of daily production.
In the legends section it warns us that there are many made up legends surrounding the Taj Mahal.
The saga of The Taj would be half told if the myths related to it are not mentioned. Like many a great buildings the Taj Mahal has its myths and legends. It seems that there is more fiction on the Taj than serious scholarly research. Several of the stories belong solely to oral tradition and are told by the guides, some are so established that they form a popular history of the monument and have made their way into guidebooks, and some have been taken up by scholars, or even created by them, and thus become part of the scholarly debate.