Haskell, 1165 1065 1053 bytes
Bytes saved thanks to Leo Tenenbaum
n=Nothing
x?y=Just(x,y)
o(x,y)=x<0||y<0||x>7||y>7
m#k@(x,y)|o k=n|1>0=m!!x!!y
z(x,y)m p(a,b)|o(x+a,y+b)=1<0|Just g<-m#(x+a,y+b)=elem g[(p,0),(5,0)]|1>0=z(x+a,y+b)m p(a,b)
t(x,y)p(a,b)m|o(x+a,y+b)=[]|g<-(x+a,y+b)=(g%p)m++do[0|m#g==n];t g p(a,b)m
c m|(x,y):_<-[(a,b)|a<-u,b<-u,m#(a,b)==6?1],k<-z(x,y)m=or$[m#(x+a,y+b)==6?0|a<-0:s,b<-0:s]++do a<-s;[k 3(a,b)|b<-s]++(k 2<$>[(a,0),(0,a)])++[m#l==4?0|b<-[2,-2],l<-[(x+a,y+b),(x+b,y+a)]]++[m#(x-1,y+a)==p?0|p<-[0,1]]
c m=1>0
(k%p)m=[[[([p|a==k]++[m#a])!!0|a<-(,)b<$>u]|b<-u]|not$o k]
w(Just(_,1))=1<0
w x=1>0
m!u@(x,y)|g<-m#u,Just(q,1)<-g,v<-((u%n)m>>=),r<-v.t u g,k<-(do[0|n==m#(x+1,y)];(u%n)m>>=(x+1,y)%g)++(do a<-s;[0|n<m#(x+1,y+a)];v$(x+1,y+a)%g)++(do[0|(x,n,n)==(1,m#(x+1,y),m#(x+2,y))];v$(x+2,y)%g)++(do a<-s;[0|1?0==m#(x,y+a)];v((x,y+a)%n)>>=(x+1,y+a)%g)=[k,k,do a<-s;[(a,0),(0,a)]>>=r,do a<-s;b<-s;r(a,b),do a<-s;b<-[2,-2];l<-[(x+a,y+b),(x+b,y+a)];v$l%g,do a<-0:s;b<-[0|a/=0]++s;r(a,b),do a<-[x-1..x+1];b<-[y-1..y+1];[0|w$m#(a,b)];v$(a,b)%g]!!q
m!u=[]
u=[0..7]
s=[1,-1]
q m=all c$m:do a<-u;b<-u;m!(a,b)
Try it online!
This is not exactly well golfed as of now, but it is a start. With some help along the way I've now golfed this down pretty aggressively (and fixed an error along the way).
The one perhaps questionable thing this does is that it assumes that, other than by a king or a pawn en passant, you can never get out of check by capturing one of your own pieces. In chess you are not allowed to make this move but my program considers these moves to save bytes under the assumption that if you are in check this can never get you out of it.
This assumption is valid because such moves
Cannot capture the piece that is attacking the king, since the piece they capture is black.
Cannot block the path of the piece that is attacking the king, since the captured black piece would have already been doing that.
We also add the additional stipulation that if you have no king you are in check.
This program also makes the assumption that if there is a pawn that can be captured en passant, then the pawn was the last piece to move and that move was a legal move. This is because the program does not check if the square it moves the black pawn to is empty so if there is a piece there things can get a little screwy. However this cannot be obtained if the last move was a legal move and furthermore cannot be represented in FEN. So this assumption seems rather solid.
Here is my "ungolfed" version for reference:
import Control.Monad
out(x,y)=x<0||y<0||x>7||y>7
at b (x,y)
|out(x,y)=Nothing
|otherwise=(b!!x)!!y
inLine (x,y) ps m (a,b)
| out (x+a,y+b) = False
| elem (m `at` (x+a,y+b)) $ Just <$> ps = True
| m `at` (x+a,y+b) == Nothing = inLine (x+a,y+b) ps m (a,b)
| otherwise = False
goLine (x,y) p (a,b)m
| out (x+a,y+b) = []
| otherwise = case m `at` (x+a,y+b) of
-- Just (n,1) -> []
Just (n,_) -> set(x+a,y+b)p m
Nothing -> set(x+a,y+b)p m ++ goLine(x+a,y+b)p(a,b)m
checkBishop (x,y) m=or[inLine(x,y)[(3,0),(5,0)]m(a,b)|a<-[1,-1],b<-[1,-1]]
checkRook (x,y) m=or$do
a<-[1,-1]
inLine(x,y)[(2,0),(5,0)]m<$>[(a,0),(0,a)]
checkKnight (x,y) m=any((==Just(4,0)).(at m))$do
a<-[1,-1]
b<-[2,-2]
[(x+a,y+b),(x+b,y+a)]
checkPawn (x,y) m=or[at m a==Just(p,0)|a<-[(x-1,y+1),(x-1,y-1)],p<-[0,1]]
checkKing (x,y) m=or[at m(a,b)==Just(6,0)|a<-[x-1..x+1],b<-[y-1..y+1]]
check m
| u:_<-[(a,b)|a<-[0..7],b<-[0..7],(m!!a)!!b==Just(6,1)] =
checkBishop u m ||
checkRook u m ||
checkKnight u m ||
checkPawn u m ||
checkKing u m
| otherwise = True
set (x,y) p m=[[[head$[p|(a,b)==(y,x)]++[(m!!b)!!a]|a<-[0..7]]|b<-[0..7]]|not$out(x,y)]
white(Just(n,0))=True
white x=False
moves m (x,y)
|g<-m `at` (x,y)=case g of
Just(2,1) -> do
a<-[1,-1]
b<-[(a,0),(0,a)]
set(x,y)Nothing m>>=goLine (x,y) g b
Just(3,1) -> do
a<-[1,-1]
b<-[1,-1]
set(x,y)Nothing m>>=goLine (x,y) g(a,b)
Just(4,1) -> do
n<-set(x,y)Nothing m
a<-[1,-1]
b<-[2,-2]
l<-[(x+a,y+b),(x+b,y+a)]
-- guard$white$n `at` l
set l g n
Just(5,1) -> do
a<-[1,-1]
c<-[(a,0),(0,a),(a,1),(a,-1)]
set(x,y)Nothing m>>=goLine (x,y) g c
Just(6,1) -> do
a<-[x-1..y+1]
b<-[x-1..y+1]
guard$white(m `at`(a,b))||Nothing==m`at`(a,b)
set(x,y)Nothing m>>=set(a,b)g
Just(n,1) -> (do
guard$Nothing==m `at` (x+1,y)
set(x,y)Nothing m>>=set(x+1,y)g) ++ (do
a<-[1,-1]
guard$white$m`at`(x+1,y+a)
set(x,y)Nothing m>>=set(x+1,y+a)g) ++ (do
guard$(x,Nothing,Nothing)==(1,m`at`(x+1,y),m`at`(x+1,y))
set(x,y)Nothing m>>=set(x+2,y)g) ++ (do
a<-[1,-1]
guard$Just(1,0)==m`at`(x,y+a)
set(x,y)Nothing m>>=set(x,y+a)Nothing>>=set(x+1,y+a)g)
_ -> []
checkmate m=all check$m:do
a<-[0..7]
b<-[0..7]
moves m(a,b)
Try it online!
