L-tyrosine is a non-essential amino acid that is produced in our bodies by breaking down another amino acid called phenylalanine.
Despite being classified as "non-essential" L Tyrosine is involved in the production of important hormones and neurotransmitters that help regulate mood, cognitive abilities, and metabolism. These include dopamine, epinephrine (adrenaline), norepinephrine, and thyroid hormones. In terms of weight loss of course, it's largely dopamine we'll be looking at.
While the body can produce L-tyrosine on its own, there have been some studies that suggest consuming extra through dietary sources or supplements can enhance energy, mood, and mental performance.
And there is some, albeit mixed evidence that it can have a positive impact on weight loss by speeding up metabolism, although a lot of this is from animal studies and the human results are mixed. In general all of the claims around it raising metabolism are based on rats, although there are some human trials that showed positive results in terms of reducing cravings, suggesting that there is something to the claims around the mood elements of L-Tyrosine.
Ultimately, there are substantially better weight loss aids than L-Tyrosine, with everything from caffeine to capsaicin having more supplorting evidence, BCAA or none essential amino supplementation is worth while in certain demographics, but most people already get more than they need.
As we mentioned, the amino acid L-Tyrosine is connected to dopamine production,  which in turn plays a crucial role in reward-motivated behavior and feelings of pleasure. Low dopamine levels have been associated with increased food cravings, overeating, and weight gain. Meaning that not having enough L-Tyrosine could have a knock on effect. But, this is ultimately where it's use case ends.
The metabolism claims tend to come as a result of thyroid hormones, and of course, for most people their thyroid's are functioning just fine. There is also the potential for people with under active thyroids to make matters worse by supplementing L-Tyrosine, meaning this is quite possibly a double edged sword. It is an essential part of producing these hormones as is iodine, but realistically this is something that should be handled by a medical proffesional.
L-Tyrosine Dosage For Weight Loss
A lot of weight loss or "fat burner" supplements contain L-Tyrosine due to the suggested metabolism impacts that we mentioned earlier. The problem is that generally doses of L-Tyrosine are quite large and not suitable for capsule supplements.
The recommended dosage for weight loss through craving suppression is a standard dose of 100-150mg per 1kg of body weight. For example, a person weighing 150 pounds would take roughly 7-10 grams of L-tyrosine, while a person weighing 200 pounds would take 9-13.5 grams. And most capsules are sized to contain 1 gram of powder. Meaning that if you're looking to use L-Tyrosine you'd want to consume it in a powder form. 
Generally speaking however, we'd recommend simply taking an amino acid powder if you think you aren't getting enough. However, most meat eaters will get more than enough of the none essential amino acid. Vegans however may have a harder time as most vegan foods do not have a complete amino acid profile (soy being the only exception), which could result in not having enough phenylalanine or L Tyrosine.
And in the case of amino deficiencies, then there is very compelling evidence to suggest supplementing would be beneficial for weight loss and in this case the results aren't nearly as mixed. There are likely more important areas for vegans to supplement ahead of amino acids like L-Tyrosine for weight los, but it's worth a consideration in this context.
So What About If You Don't Have Enough L Tyrosine
L Tyrosine effects the neurotransmitters dopamine and serotonin. These do play a role in the regulation of energy metabolism, feeding behavior, and physical activity. This is where. Studies have shown that tyrosine supplementation can increase dopamine levels in the brain, leading to improved mood, focus, and physical performance. These effects may indirectly promote weight loss by increasing motivation for exercise and reducing emotional eating.
Some example studies
In a study conducted on mice, the addition of tyrosine to a high-fat, high-carbohydrate diet (HFCD) resulted in increased anxiety and muscle tone. This suggests that tyrosine may have a modulating effect on the dopamine neurotransmitter metabolism, leading to changes in behavior and muscle activity. 
A study conducted on overweight women found that L-tyrosine supplementation reduced food cravings and increased feelings of fullness, leading to a decrease in calorie intake. These findings suggest that L-tyrosine may be beneficial for individuals looking to control their appetite and reduce calorie consumption. 
Is L-Tyrosine Good For Weight Loss?
Well, if you have a deficiency sure, as not having enough can reduce dopamine, seratonin regulation, and some thyroid hormones. If you don't then it's probably not going to do all too much, it certainly isn't going to "burn" a significant amount of fat. There's a bit of evidence that it can help with cravings and mood, but we wouldn't recommend looking for it as an individual supplement. Really this one's probably just for vegans in which case getting an amino acid supplement could have some value.
1 - https://pubmed.ncbi.nlm.nih.gov/1359904/
2 - https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1863555/
3 - https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5412138/
4 - https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5412138/