If one wants to make the uni lighter by putting on thin racing tyres,
will the bike become harder to handle (on paved surface)?
whats this bike u speak of?
If you’re going to save weight on the uni, then the tyre may be the worst place to start.
I have a Nimbus 28 with the original 700c x 32 mm road tyre, and 110mm cranks. It’s light, nimble, and great fun to ride, but I think it’d be faster over a journey if it had a heavier, fatter 29 inch tyre.
A heavy tyre (or rim) adds flywheel mass to the wheel. It will make the uni slightly harder to acclerate (less responsive, slightly trickier to idle) but the converse is that if it’s harder to accelerate, it’s also harder to decelerate, and it will take less notice of minor irregularities in the road surface. It will be easier to ride fast.
One of my pipe dreams is to have a super lightweight 28 with an ultra narrow road rim. It’ll look good, be fun to ride, but be totally impractical on anything but super smooth concrete. Rogeratunicycledotukdotcom strongly advised me against the skinny lightweight option. (I’ll probably still do it one day, though ;o) )
So, for speed, go for a forgiving tyre, rather than a skinny lightweight one.
Now, look at weight generally. Why do you want to save weight on your unicycle?
I weigh approximately 150 pounds. My unicycles weigh less than 10% of that. A litre of water weighs about 1.75 pounds, which is over 10% of the weight of the unicycle. You can shave grammes off the weight of a unicycle, but they are a tiny percentage of the total weight that you are trying to propel.
An experiment was done with bicycles about 20 years ago, and was reported over here in the bicycle press. A time trials rider rode 25 miles, then repeated the ride a few days later in virtually identical conditions, but with 3 pounds of lead attached to the frame of his bike. It made seconds’ difference over the 25 mile - not enough to draw any conclusion except that the weight made no appreciable difference.
Mass resists acceleration and deceleration. Get a heavy wheel, or unicycle, moving, and it will be harder to stop. You get the famous flywheel effect beloved of Coker riders.
It is worth saving weight if you’re going to hop, drop, jump, or climb, but then you would be choosing a tyre appropriate for those activities, rather than trying to get a light tyre.
The flywheel effect may be great on a Coker, but on a smaller wheel its just annoying. I had an old 28" uni with a heavy steel rim, heavyweight tyre and thickish tube. It was a pig to ride. One of the easiest ways to make your uni lighter is to get a lighter wheelset. By wheelset I mean lighter hub, spokes, nipples, rim, tube, rim strip and tyre.
An ultra skinny road tyre is great if you plan on road and track riding only. For other riding it will suck.
Re: chasing grams
Thanks for some good answers! I should have been a little more elaborate
though: My thought on chasing grams came from envisioning myself doing a
longer hike combining unicycling with walking/carrying the uni.
I agree with you Mikefule that in enhancing (any biking) performance you
might as well start shaving off weight at other places than on the bike
itself. Hm, I could start with my belly 
GC
Maybe the bike that the tyres have come off?? 
In that case, the bike would be much harder to handle. Have you ever tried riding a bike without tires?