Burke was first elected to the 7th District of the Erie County Legislature in 2013. In August 2015, Burke sponsored a bill that became the first microbead ban in the state of New York. It bans the sale and distribution of all plastic microbeads including from personal care products., its prohibition on sales is stronger than any other law in the country. It was enacted on August 12, 2015 and took effect in February, 2016. In November 2015, four other NY counties followed suit. In 2015, Burke released a plan to create a regional municipal broadband network as co-chair of the Erie County Municipal Broadband Project Committee. The report found that affordable, high-speed municipal broadband internet service in Erie County would create faster job and business growth, while raising property values. Burke introduced the Prevention of Emotional Neglect and Childhood Endangerment Act, named for Vice President Mike Pence. The law bans conversion therapy, and went into effect upon filing with the New York Department of State on March 7, 2018.
Burke won the November 6, 2018 election for the 142nd district of the New York State Assembly. During the 203rd New York State Legislature, Burke co-sponsored laws that allow early voting, no-excuse absentee voting, and regulate political donations by Limited Liability Companies. In 2018, Burke called on Diocese of Buffalo Bishop Richard Joseph Malone to resign over accusations of negligence concerning investigations into priests accused of sexual abuse. On January 28, 2019, Burke co-sponsored the Child Victims Act that allows prosecutors to bring criminal charges until a victim turns 28, and permits victims to sue until age 55. In April 2019, Burke introduced and passed legislation that permits the New York Office of Victim Services to provide compensation and funding to a broader classification of child victims, for uses such as mental health counseling and transportation to court appearances. In February 2019, Burke joined his fellow Western New York elected officials in condemning Governor Cuomo's budget that cut state Aid and Incentives Municipalities funding for nearly all towns and villages. Burke cosponsored Assembly Bill A576, which passed the State Assembly on January 15, 2019 by a vote of 141-7, and passed the State Senate that same day by a vote of 57-4. The bill expanded the Erie County conversion therapy ban to cover the entire state, and was signed by Governor Andrew Cuomo on January 25, 2019, taking effect immediately upon receiving his signature. Since 2016, conversion therapy has been banned in New York from coverage by insurers and Medicaid, or from being practiced by any state-licensed or operated mental health facility per executive order.
