The saffron party leaders held a dharna near BJP state headquarters here, when Patnaik was addressing school students on the occasion of Children's Day.
The BJP also launched a website for children to highlight the "failures" of the state government during its 17-year rule.
"The state's BJD government has been misusing tax-payers' money to gain political mileage for chief minister Naveen Patnaik," BJP state general secretary P Harichandan alleged.
The party also criticised Patnaik's mode of addressing children earlier in the day.
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Taking a jibe at him, BJP vice-president Sameer Mohanty said, "How could Naveen Patnaik suddenly develop love for children after 17 years of rule? The hidden reason is publicity using public money."
Mohanty further alleged that the state government spent about Rs 200 crore from the state exchequer for Children's Day celebrations.
"The state government doesn't need the permission from BJP to undertake any expenditure for children," said BJD spokesman Samir Dash.
"If the CM wants to give students a message on Children's Day, the BJP should be happy. Why it is getting so intolerant?" Dash wondered.
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