You need to specify the file name, since the default date will contain special characters:
lynx -dump http://somesite/script.php > "$(date).txt"
As @Gareth says, you should specify a format string for the date so that you get a more readable / manageable file name, e.g.
lynx -dump http://somesite/script.php > "$(date +%Y%m%d-%H%M%S).txt"
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