Three Conservative MPs have stated that they intend to resign as MPs "
with immediate effect."
The most notable of them is former Prime Minister Boris Johnson (Uxbridge and Ruislip South).
The other two are Nadine Dorries (mid Bedfordshire) and Nigel Adams (Selby and Ainsty).
In the December 2019 General Election, Johnson won his seat with 25,351 votes as against Labour's candidate (18,141), the Liberal Democrat (3026) and 9 other candidates. In the current state of national politics, this seat appears to be winnable by Labour.
Dorries won an enormous majority - 38,692 votes to Labour (14,028), Liberal Democrat (8171), Green (2478) and two other candidates. Adams secured a large majority with 33,995 votes to Labour (13,858), Liberal Democraft (4842), the Yorkshire Party (1900) and Green (1823). Perhaps those seats will remain with the Conservatives (with reduced majorities) but Labour will certainly be hoping to win the seats.
An interesting fact is that Under a Resolution of the House of Commons dated 2 March 1624, Members of Parliament cannot directly resign their seat. A sitting MP has to apply to the Chancellor of the Exchequer for one of two ancient "offices" - either Crown Steward and Bailiff of the Chiltern Hundreds or Crown Steward and Bailiff of the Manor of Northstead. The process is described at -
Resignation from the House of Commons - House of Commons Library (parliament.uk)
It is possible that the Chancellor might refuse an application but that is unlikely and has not actually happened since 1842. The last person to be refused either office was Viscount Chelsea, who
was refused the Chiltern Hundreds in 1842. This was due to suspicions
that money had changed hands to influence the result of the by-election which would have followed his appointment.
Once an MP has gone, a by-election usually follows but that cannot happen until a WRIT is moved by the House of Commons. By-elections are explained at - By-elections - UK Parliament
A motion to issue a writ is left to the Chief Whip of the party whose MP held the now vacant seat.
Occasionally, Members have resigned their seats and then fought the
ensuing by-election.
State of the parties - MPs and Lords - UK Parliament
Note 12 June 2023:
Three Hundreds of Chiltern - GOV.UK (www.gov.uk) - Johnson appointed Steward.
Manor of Northstead - GOV.UK (www.gov.uk) - Adams appointed Steward