No; you do not pay.
From the Official Rules of Monopoly on the Hasbro web site under heading Mortgages (my emphasis):
No rent can be collected on mortgaged properties or utilities, but rent can be collected on unmortgaged properties in the same group.
To me this is unequivocal - no payment, whether or not denoted as rent, is ever collected on a property that is face down due to being mortgaged. Not carefully the wording "in the same group", and that it does not say "in the same colour group"; thus clearly indicating an applicability to railroads as well as coloured properties.
While an omission from the rules may, depending on circumstance, be regarded as an oversight, I can think of no reason why an explicit inclusion in the rules should not always be taken at face value. Clearly there is an omission about whether or not rent includes payment for landing on a Utility. However the explicit exclusion of payment for landing on a mortgaged utility is clearly an intentional inclusion - so payment is not due under this circumstance.
As the rules never make any distinction on how one lands on a square, to suppose that there is intended to be one for advancing to a Utility in consequence to drawing a Chance or Community Chest card is completely unreasonable.
Update
The claim is being made in comments that
"Your answer seems to be based on an implicit assumption that the payment demanded by the card is rent, or that when that rule says "rent" it means "rent or other payments", but it doesn't say that and neither do the rules."
I submit as counter argument this quote, again from the Official Rules of Monopoly.
Paying Rent:
When you land on property owned by another player, the owner collects rent in accordance with the list provided on the Title Deed Card.
Note the clear implication (at least), that the payments collected in consequence to landing on a property owned by another player are properly termed "rent".
Update - from the Official Tournament Guide for Monopoly, Page 7
Paying Rent on a Utility you throw the dice and land on Water Works or the Electric Company and another player owns them, don’t touch the dice. See what the total is. Multiply that number by 4 if your opponent only owns one utility, or by 10 if he/she owns both of them.
Paying Rent on a Fully Owned Property Group Suppose your opponent owns a complete color-group — for example, all three reds — with no buildings on any of them. If you land on one of the reds, you have to pay double its rent. But suppose one of the reds — e.g. Indiana Avenue — is mortgaged and you land on Illinois, it is unmortgaged. How much rent do you pay? Double, even though Indiana is mortgaged, your opponent still owns it.
Note that this Official guide explicitly refers to the amount paid for landing on a utility as rent. And the Rules otherwise clearly state that rent is not paid for landing on mortgaged property.