Cases where confusion can arise:

1. One day mom buys new flowers, and places them in a vase near the dining table. During dinner, her son says to her "Hana ga kirei dane"(the flowers are pretty), and she mistakenly became very happy about her son.

2. Grandmother says "takusan kumo ga detekita" (lots of clouds are coming out) to which her grandson paranoidly responds "where? when?"

3. Stranger calls house, little kid picks up. Stranger says "okaasan iru?" (Is your mom there?) The child responds: "iranai!" (no thank you!)

Some homophones:

dekiru - has skill (ex: "kare wa dekiru" = "he's got skill")
dekiru - to be able to
dekiru - to conceive a life form
furu - to fall, pour ("ame ga furu" == "rain falling")
furu - to shake
haku - throw up
haku - to put on (ex: "kutsu wo haku" = "to put on shoes")
hana - flower
hana - nose
iru - at/exist
iru - need/want
kaburu - to get a rash
kaburu - to wear (ex: "booshi wo kaburu" = "to wear a hat")
kaki* - mussel
kaki* - persimmon
kaku - to itch
kaku - to write
kiku - chrysanthemum
kiku - to be effective
kiku - to listen to
kiru - to cut
kiru - to wear (ex: "T-shatsu wo kiru" = "to wear a T-shirt")
kumo - cloud
kumo - spider
sake - rice wine
sake - salmon fish (syn: shahke)
tatsu* - dragon
tatsu* - to stand
umi - ocean, sea
umi - oozing puss

re Broccolist: True, chizu and chiizu are not homophones, and the difference should be discernable to a speaker of almost any language upon hearing. There are pseudo-homophones that have tone difference(Japanese accent), as you point out. The fake ones are marked with asterisks.