Leaders from Muhammadiyah, Indonesia's second biggest mass Muslim organisation, also urged the government to revoke the company's business licence for its stance on lesbian, gay, bisexual and transgender (LGBT) issues.
"The ideology, business and view that they support are against our ideology," Anwar Abbas, Muhammadiyah's head of economic affairs, told AFP.
Another Muhammadiyah leader, Yunahar Ilyas, said the group was calling on "Muslims to not drink in Starbucks so that the income is not used to strengthen LGBT campaigns".
The firm is among a slew of US companies to have spoken up against discrimination in the US, with representatives signing a letter to North Carolina's Republican governor protesting legislation targeting transgender people last year.
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Hardliners and Islamic groups have led a growing backlash against Indonesia's LGBT community over the past year, which activists believe was triggered by widespread media coverage of a decision in the United States to legalise same-sex marriage.
The group Pribumi Perkasa Malaysia this week also called on the government to "re-evaluate the trading licence given to companies that support same-sex marriages and LGBT," spokesman Amini Amir Abdullah said in a statement.
At the time, chairman Howard Schultzman, then the chief executive, responded by asking the shareholder to sell the shares.
Homosexuality has long been taboo in Malaysia, where 60 percent of the population is Muslim, and where sodomy is a crime punishable by up to 20 years in prison.