Interesting use case!! (difficult to understand in first reading)
If you want to see only "duplicated" rows , dplyr's n() in summarise gives count of every grouped variable(s) and you can filter any with more than 1 value:
df %>%
  group_by(Longitude, Latitude) %>%
  summarise(N=n())%>%filter(N>1)
And if you want the summation of Prob column , then as suggested by Merijn van Tilborg simply sum(Prob).
But if u want what's mentioned in last line of your post, then slightly more work needed->
your question has more than just appears (slightly ambiguous too). With "AND of only to merge where Prob is 1 and 2"
"If two rows have the same longitude and latitude values, and if they show Prob 1 and 2, I want to "merge" these rows and make Prob 3."
then we need to combine the two:
some data similar to yours:
dx <- data.table(
  long = c('a','b','d1','c','a','d1','e','f','a')
  ,lat = c('a','b1','d1','c','a','d1','e','f','g')
  ,Prob = c(1,0,2,1,0,1,0)
9 records
  long lat Prob
1:    a   a    1
2:    b  b1    0
3:   d1  d1    2
4:    c   c    1
5:    a   a    0
6:   d1  d1    1
7:    e   e    0
8:    f   f    1
9:    a   g    0
Now in 9 records 1st and 5th are same but different Probs; Not 1 AND 2
but 3rd and 6th have 1 AND 2, which can be filtered as !=0;
So you want to merger only 3rd and 6th BUT not 1st and 5th and hence output shud have 8 records !
1st approach : with error (if you merge commons-> then it would be 7 with simple sum of Prob!=0 like this)
dx%>%
  group_by(long,lat)%>%
  summarise(N=n()
            ,Probs = sum(Prob[Prob!=0])
            )
yields 7 records merging 1st and 5th (a,a):
  long  lat       N Probs
  <chr> <chr> <int> <dbl>
1 a     a         2     1
2 a     g         1     0
3 b     b1        1     0
4 c     c         1     1
5 d1    d1        2     3
6 e     e         1     0
7 f     f         1     1
But if you dont want to do that:
then grouping may not be required. (Generally that's first thing  comes to mind for dplyr users like me)
2nd idea (it works) :
get the data with Prob==3 first and then add it to original data after removing the eligible rows.
p3 <- dx%>%
  group_by(long,lat)%>%
  summarise(Prob = sum(Prob[Prob!=0])
            )%>%filter(Prob==3)
yields
  long  lat    Prob
  <chr> <chr> <dbl>
1 d1    d1        3
now your desired output :
dx%>%
  arrange(long)%>%
  filter(!(long %in% p3$long & lat %in% p3$lat))%>%
  bind_rows(p3%>%select(long,lat,Prob))
as
   long lat Prob
1:    a   a    1
2:    a   a    0
3:    a   g    0
4:    b  b1    0
5:    c   c    1
6:    e   e    0
7:    f   f    1
8:   d1  d1    3
please try and let me know if it works.