Burns had not competed since he passed out at the wheel of his road car on the way to the Wales Rally GB in 2003.
He was diagnosed with a brain tumour and underwent surgery in April.
Burns was championship runner-up in 1999 and 2000 before becoming the first Englishman to land the coveted world title 12 months later.
Burns' family paid tribute to him in a statement, saying: "From the outset Richard knew that the odds were heavily against him and yet he fought his illness with bravery and good humour.
"Having undergone both chemotherapy and radiotherapy he was able to leave hospital in summer 2004.
"For a while his health showed signs of improvement but then after six months it once again began to decline.
"Determined not to give up, he opted for surgery earlier this year. This alleviated some of the symptoms of his illness and enabled him to remain active.
"At Castle Combe in August he attended a parade of the rally cars that he drove throughout his career and was touched by the warmth of the reception he received.
"However there was to be no miracle and in recent days he lapsed into a coma.
"The date of a memorial service will be announced shortly."
RICHARD BURNS FACTFILE Born: 17/1/1971, Reading Rallying debut: 1988 World Championship 1998: First win at Safari Rally, also wins Rally of Great Britain World Championship 2001: Wins world title for Subaru, 1st Englishman to do so
Burns made his rallying breakthrough in 1990 when he won the national 205GTI challenge series and he then lifted the Mintex National series title.
He then joined Subaru and became the British Championship's youngest winner in 1993.
After a spell in the Asia Pacific Championship and the occasional world championship drive, he entered for his first full season in 1998, partnering world champion Tommi Makinen for Mitsubishi.
Twelve months later he moved to Subaru and he made his first title challenge as wins in Greece, Australia and Britain helped him finish second in the final standings.
In 2000 he looked on course to win the world title, having led the championship race for some time, but he was pipped to glory by Marcus Gronholm - even though he won the season-ending Rally of Great Britain.
However he was not to be denied and in 2001 he became the first Englishman to take the championship.
After a poor start to the season, he finished second behind Colin McRae in Argentina to spark his challenge. Two more runners-up spots followed before he earned his first victory of the campaign in New Zealand.
Another second spot - this time in Australia - set up a thrilling finale at the Rally of Great Britain where Burns delighted his home fans by finishing third, which was enough to clinch the title.
He switched teams in the close-season, leaving Subaru for Peugeot but did not manage to win an event all season despite claiming four second-place finishes as he came fifth in the championship standings.
Burns made a good start to his 2003 campaign and he led the field by six points after three events and held a four-point lead in the middle of the season.
He lost his lead with two events to go and was taken ill on his way to the season-ending Rally GB.
Brytyjski kierowca rajdowy Richard Burns zmarł w piątek w wieku 34 lat po długiej chorobie. Dokładnie cztery lata temu, w 2001 roku, wygrał zawody World Rally Championship.

Kłopoty zdrowotne Richarda Burnsa rozpoczęły się do rajdu Walii na Mistrzostwach Wielkiej Brytanii w 2003 roku. Podczas drogi na rajd zemdlał za kierownicą swojego samochodu. Od tamtej pory nie brał udziału w zawodach rajdowych.

Przeprowadzona diagnoza wskazała na guza mózgu. W kwietniu przeszedł operację. Potem poddawał się zabiegom chemio- i radioterapii. Latem 2004 roku, gdy jego stan zdrowia uległ poprawie, opuścił szpital. Jednak po sześciu miesiącach zaczął czuć się coraz gorzej. W ostatnich dniach zapadł w śpiączkę.

Data pogrzebu zostanie niebawem zapowiedziana.

Richard Burns urodził się 17 stycznia 1971 roku w Reading, w Wielkiej Brytanii. Jego pierwszy kontakt z samochodem miał miejsce w wieku ośmiu lat. Wtedy jego ojciec, Alex, pozwolił mu pokierować samochodem marki Triumph w polach w pobliżu ich domu. 

Pierwszy raz za kierownicą rajdowego samochodu zasiadł w 1988 roku. Dziesięć lat później odniósł pierwsze zwycięstwo w jednym z rajdów World Rally Championship. W 1999 roku zajął drugie miejsce w klasyfikacji końcowej WRC, zaś w 2001 wygrał całe zawody. Był pierwszym Brytyjczykiem, któremu udała się ta sztuka. Startował wówczas z ekipą Subaru.