8 yr old diagnosed with sinus arrhythmia. How high of a heart rate is too high? His heart rate is 150-165 BPM just putting shoes and coat on. He is alays out of breath. Diagnosed asthma but wondering.

Check thyroid. Possibly from an overactive or cardiac issue(mitral valve). Need to see a cardiologist if asthma is under control.
Too high. A sustained heart rate of 150-165 with activities of daily living is too high and needs to be evaluated and treated asap. Assure he gets to see a pediatric cardiologist for a medical evaluation and the proper treatment. Best wishes to you.