The Hammersmith and City Line has been suspended between Edgware Road and Hammersmith over fears debris could fall on to the tracks.
The Circle Line is also closed partly due to the fire and partly because of planned engineering work.
Transport for London (TfL) said the move had been made "at the request of the London Fire Brigade".
The fire broke out at the 24-storey block, which contained 120 one and two-bedroom flats, early on Wednesday morning.
Police presume 58 people have died but have warned the figure could grow.
A sign at Ladbroke Grove station said "owing to the safety of the Tower" the lines had been partly suspended.
The London Fire Brigade confirmed it had requested the lines be temporarily closed because of a "short-term risk of some debris falling on to the tracks".
A spokesman said teams were working to secure the debris so the lines could be reopened as soon as possible.
Two London Tube lines remained part suspended on Sunday due to safety concerns with Grenfell Tower.