The TARDIS is not unusually heavy, although no precise figure for its weight has ever been given on screen. But it may weight no more than a few ounces.
During the Jon Pertwee era, because it was 'grounded' for many years owing to the Doctor's exile on Earth, UNIT were occasionally compelled to move the TARDIS from place to place by the simple expedient of brute force, and always managed to do so without any obvious difficulty. All that appeared to be needed was a squad of soldiers - and possibly a rope.
In Spearhead From Space (1970), the Brigadier had the recently landed TARDIS moved by UNIT from its landing site in Oxley Woods near Ashbridge to UNIT headquarters in London.
In The Claws of Axos (1971), the Brigadier had the TARDIS moved, by lorry, from UNIT HQ to the Nunton nuclear power complex in Southern England.
In The Time Monster (1972), Captain Yates and a squad of soldiers moved the TARDIS by lorry from UNIT HQ in London to the TOMTIT facility at Wooton outside Cambridge.
More unusually, in Carnival of Monsters (1973) the Lurman carnival proprietor, Vorg, was able to hold the miniaturised TARDIS in one hand when showing it to his assistant, Shirna, on discovering it inside the miniscope (establishing that, on this occasion at least, its weight was not more than a few ounces).
As has already been mentioned, in earlier serials the TARDIS had been moved around on a number of occasions. An early such incident took place in the Hartnell serial The Web Planet (1964), when the Zarbi dragged it away across the plain of Vortis. However, the very earliest was in Marco Polo (also 1964), when Polo's chinese servants carried the TARDIS from the Roof of the World to Peking on a handcart.
Also not mentioned previously is that the TARDIS is so light that when it landed on the sea during the Troughton serial Fury from the Deep (1968) it floated, and the Doctor and his companions were obliged to paddle ashore in a rubber dingy.
Other occasions include the time in Egypt when the slaves of Pharaoh manhandled it a few dozen yards from the open desert into the interior of a nearby Pyramid, during the Hartnell serial The Daleks Master Plan (1965); and when it was stolen from Gatwick Airport in London in the first episode of the Troughton story Evil of the Daleks (1967) and driven away on the back of a truck - on which occasion the theft appeared to be managed by just two men.
Subsequently, in the Tom Baker serial Logopolis (1981) we witnessed the partially miniaturised TARDIS being carried through the streets of Logopolis - again by just two men.
There were, now I come to think of it, many similar incidents. For instance, in The Faceless Ones (1967) the TARDIS landed in the middle of the main runway at Gatwick Airport, and had to be moved by the airport police. And while visiting London's Heathrow Airport, in Time Flight (1982), the TARDIS was loaded onto a Concorde for the flight into the past using only a small fork-lift truck.
Accordingly, the suggestion that the TARDIS weighs millions of pounds is simply not born out by the evidence.