Checking if points fall within polygon Shapefile

The spatial data.frame is not correctly formed. This might work:

library(rgeos)
library(sp)
library(rgdal)

wa.map <- readOGR("ZillowNeighborhoods-WA.shp", layer="ZillowNeighborhoods-WA")

sodo <- wa.map[wa.map$CITY == "Seattle"  & wa.map$NAME == "Industrial District", ]

# Don't use df as name, it is an R function
# Better to set longitudes as the first column and latitudes as the second
dat <- data.frame(Longitude = c(-122.332271,-122.353985,-122.331639),
                  Latitude =c(47.591351, 47.62212,47.595152),
                  names = c("Safeco Field", "Key Arena", "Century Link"))
# Assignment modified according
coordinates(dat) <- ~ Longitude + Latitude
# Set the projection of the SpatialPointsDataFrame using the projection of the shapefile
proj4string(dat) <- proj4string(sodo)

over(dat, sodo)
#  STATE COUNTY    CITY                NAME REGIONID
#1    WA   King Seattle Industrial District   271892
#2  <NA>   <NA>    <NA>                <NA>       NA
#3  <NA>   <NA>    <NA>                <NA>       NA

over(sodo, dat)
#           names
#122 Safeco Field

I've just been doing the same thing. Pascal's answer almost covers it but you may need two extra steps as below.

#After you create your list of latlongs you must set the proj4string to longlat
proj4string(dat) <- CRS("+proj=longlat")

#Before you re-set the proj4string to the one from sodo you must actually convert #your coordinates to the new projection
dat <- spTransform(dat, proj4string(sodo))

I used a similar approach to the accepted answer in this post but was never really satisfied with it so I looked into alternatives and found the sf library.

And using this library you can then write code like this:

library(sf)
# Shapefile from ABS: 
# https://www.abs.gov.au/AUSSTATS/[email protected]/DetailsPage/1270.0.55.004July%202016?OpenDocument
map = read_sf("data/ABS/shapes/SUA_2016_AUST.shp")

pnts_sf <- st_as_sf(pnts, coords = c('y', 'x'), crs = st_crs(map))

pnts <- pnts_sf %>% mutate(
  intersection = as.integer(st_intersects(geometry, map))
  , area = if_else(is.na(intersection), '', map$SUA_NAME16[intersection])
) 

pnts

Output:

         geometry intersection area    
*     <POINT [°]>        <int> <chr>   
1 (138.62 -34.92)           79 Adelaide
2 (138.58 -34.93)           79 Adelaide
3 (138.52 -34.95)           79 Adelaide
4 (152.71 -27.63)           60 Brisbane
5 (153.01 -27.57)           60 Brisbane
6  (150.73 -33.9)           31 Sydney  
7 (150.99 -33.92)           31 Sydney 

I posted this code on another post which was a similar question, here: Identify polygon containing point with R sf package