Is there a similar English phrase for this Tamil proverb - "Lavish outside home yet starving inside of it"?

Not an exact match but you might consider keeping up appearances.

From dictionary.cambridge.org

to ​pretend to be ​happier, less ​poor, etc. than you really are, because you do not ​want ​people to ​know how ​bad ​your ​situation is:

The previous answers are reasonable, but rather polite. British English has a rude, but also rather funny, expression which has much the same meaning: "all fur coat and no knickers".

The outside view (fur coat) is fine and expensive, but the wearer of the fur coat can't afford any underwear.

The expression isn't very common and perhaps slightly old fashioned but I would guess most Brits would know it.

ETA: To be fair, as pointed out below, the implication here isn't strictly financial it's more that a posh appearance hides a crude reality.