This is discussed in Mishna Chagiga 3:8.
As seen below, most keilim could become tamei, at which point they would be subject to same purification process as a person -- sprinkling of ashes and mikvah.
כיצד מעבירים על טהרת עזרה, מטבילין את הכלים שהיו במקדש, ואומרין להם, הזהרו שלא תגעו בשלחן (ובמנורה) ותטמאוהו .
כל הכלים שהיו במקדש, יש להם שניים ושלישים, שאם נטמאו הראשונים, יביאו שניים תחתיהן.
כל הכלים שהיו במקדש, טעונין טבילה, חוץ ממזבח הזהב ומזבח הנחשת, מפני שהן כקרקע, דברי רבי אליעזר.
וחכמים אומרים, מפני שהן מצפין.
How did they remove [the vessels] for the purification of the Temple Courtyard? They would immerse the vessels that were in the Temple, and they would say to them [the priests to the people]: be careful to not touch the table and make it impure. For each of the vessels that were in the Temple, there were two or three sets, so that if the first vessels were rendered impure, they would bring the second in their place. All the vessels in the Temple required immersion, except for the golden altar and the bronze altar, because they were like [i.e., affixed to] the ground [and thus not susceptible to impurity]; the opinion of Rabbi Eliezer. However, the sages say [they were not susceptible to impurity] because they were plated. (Translation Sefaria)