Hey, on forums like this, it is usually a good idea to provide code that can produce your input data. Then, people will have less time to spend in tackling your problem.

To do what you want, you should use GenomicRanges:

create your data

SNPs <- data.frame(
  Chr = c(1,1,2,2,3,3,4),
  POS = c(2,3,4,7,5,13,13))
SNPs
  Chr POS
1   1   2
2   1   3
3   2   4
4   2   7
5   3   5
6   3  13
7   4  13

genes <- data.frame(
  Chr = c(1,2,2,3,3,4),
  start = c(2,3,6,4,11,14),
  end = c(3,5,10,10,17,17),
  id = c('a','b','c','d','e','f'))
genes
  Chr start end id
1   1     2   3  a
2   2     3   5  b
3   2     6  10  c
4   3     4  10  d
5   3    11  17  e
6   4    14  17  f

convert the data to GRanges objects

require(GenomicRanges)
grSNPs <- makeGRangesFromDataFrame(
  SNPs,
  seqnames.field = 'Chr',
  start.field = 'POS',
  end.field = 'POS',
  keep.extra.columns = TRUE)
grgenes <- makeGRangesFromDataFrame(
  genes,
  seqnames.field = 'Chr',
  start.field = 'start',
  end.field = 'end',
  keep.extra.columns = TRUE)

find overlaps and save the indices that overlapped

qHits <- queryHits(findOverlaps(query = grSNPs, subject = grgenes, type = 'within'))
subHits <- subjectHits(findOverlaps(query = grSNPs, subject = grgenes, type = 'within'))

use indices to produce the original data for the overlaps

overlaps <- data.frame(SNPs[qHits,], genes[subHits,'id'])
colnames(overlaps) <- c('Chr','POS','id')

now find those that didn't overlap

missing <- SNPs[-qHits,]
nmissing <- nrow(missing)
missing <- data.frame(missing, rep(NA, nmissing))
colnames(missing) <- c('Chr','POS','id')

rbind overlapping regions and non-overlapping

finalresult <- rbind(overlaps, missing)
finalresult
  Chr POS   id
1   1   2    a
2   1   3    a
3   2   4    b
4   2   7    c
5   3   5    d
6   3  13    e
7   4  13 <NA>

Using data.table:

# example data (partly stolen from Kevin's answer)
SNPs <- data.table(
  Chr = c(1,1,2,2,3,3,4),
  start = c(2,3,4,7,5,13,13),
  end = c(2,3,4,7,5,13,13),
  key = c("Chr", "start", "end"))

genes <- data.table(
  Chr = c(1,2,2,3,3,4),
  start = c(2,3,6,4,11,14),
  end = c(3,5,10,10,17,17),
  id = c('a','b','c','d','e','f'),
  key = c("Chr", "start", "end"))

library(data.table)

foverlaps(SNPs, genes)
#    Chr start end   id i.start i.end
# 1:   1     2   3    a       2     2
# 2:   1     2   3    a       3     3
# 3:   2     3   5    b       4     4
# 4:   2     6  10    c       7     7
# 5:   3     4  10    d       5     5
# 6:   3    11  17    e      13    13
# 7:   4    NA  NA <NA>      13    13