Because dilute urine is usually a sign of chronic renal failure.

Hyperthyroidism "speeds" up almost all of the systems in the body, including
the rate at which the kidneys filter the blood (RBF/GFR [glomerular
filtration rate] ).  This can mask chronic renal failure in cats with
underlying disease.  After the cat's hyperthyroidism is treated and renal
blood flow and GFR return to their normal rates, toxic waste products can
build up in the blood because the failing kidneys are unable to filter out
the waste products as fast as they're produced.

A "galloping heartbeat" is a common finding in cats with hypertrophic
cardiomyopathy (HCM).  Hyperthyroidism causes myocardial hypertrophy that's
similar to HCM.  However, after hyperthyroidism is treated, hypertrophy
usually resolves

You might want to ask your vet to check your cat's blood pressure.  Since
the advent of noninvasive blood pressure monitoring techniques, many
hyperthyroid cats have concurrent hypertension -- which also often resolves
when the hyperthyroidism is treated.

As far as treatment, I strongly suggest you try oral medication first and
monitor kidney function for a month or two to see if kidney function
declines.  I would not recomment radioiodine tx for a 16 year old cat due to
the high probably of underlying CRF.

If your cat cannot tolerate Tapazole, speak to your vet about carbimazole -
it produces fewer and milder adverse effects.  However,  carbimazole is only
availabe through compounding pharmacies in the US.

Best of luck.

Phil

Are her stools soft?

Dilute urine is not bad in itself, the cause for dilute urine is what
may be bad. Dilute urine can be caused by an increased intake of
liquids (e.g., a canned diet or drinking milk regularly) or it can be
because of kidney failure. It can also be due to increased water
intake because of diabetes, hyperthyroidism or diet (e.g., eating too
much salt).

Hyperthyroidsm accelerates basal metabolism. Cells working faster
require more oxygen. This calls for an accelerated heartbeat to supply
the elevated oxygen demand. Red blood cell concentration also
increases to aid in the transport of this extra demand.

http://www.vet.purdue.edu/vcs/scottmon/hyperthycat.html

Thanks for the info.