Do the "big cats" such as tigers and lions purr like household cats?

NO; BIG CATS CAN ONLY PURR WHEN BREATING OUT.
There are five species in the genus Panthera: tigers (P. tigris), snow leopards (P. uncia), leopards (P. pardus), jaguars (P. onca), and lions (P. leo).
All Panthera cats have elastic sections on both sides of the hyoid bone, a structure which supports the tongue and its muscles.
The elastic hyoid, combined with the fibroelastic tissue on top of the big cats' undivided vocal folds, acts like a slide trombone, enabling the big cats to roar. Only snow leopards, which lack the specialized vocal folds, are not able to roar. Because of the elastic hyoid, big cats can purr only when breathing out. The hyoid of smaller cats is solid bone. These cats can purr when breathing both in and out, but they can't roar.

No they don't purr like our furry friends the cat/cats

Cheetahs can purr :) but most wild cats can not purr.

Suuuure do

Yes. Jay Leno was playing with a tiger on his shows when he started purring.

Cheetahs and cougars (aka mountain lions) can purr. And boy are they LOUD when they do! They are not in the panthera genus, I'll admit, but they nonetheless get qualified as 'big cats' in my book. As I understand it, tigers can't purr breathing out either; the closest they come is the chuffing sound they make to express affection. Some people take this to be a purr. Bobcats and lynxes can purr as well, though their 'big cat' status isn't so warranted. Medium-sized cats maybe. I can tell you that I once had two rabbits that learned to purr just by being around a housecat. Those were some unusual rabbits though. Very smart as rabbits go. So who knows. Maybe the big cats could purr if they wanted to and were brought up around some cats that are more well-known for purring, but they just don't because it doesn't come naturally to them or without having done it a lot since their youth, it's too hard on their throats (so like learning an instrument - you're only good at it if you started young). It's hard to guess at their motivations, if when they don't do something, whether it means that they can't or just don't want to. It is widely believed that no members of the cat family are able to taste sweetness (which makes them a lot less likely to poison themselves licking up antifreeze like dogs), but people only arrived at that conclusion by observing that they appear to show no preference for sweetened foods, for or against. Can they really not taste it, or are they just stoicly indifferent? We need a talking cat to tell us the answer to that one. And a talking tiger to tell us once and for all if he can't purr or if he just doesn't want to.

Yes

Big cats, such as lions and tigers, can not purr. On the other hand, small cats like bobcats and leopards, can purr but they can not roar.