I had the same problem. I am using multisite installation and the website with weird icons is secondary site (mia.handysvandy.net) with different domain as main domain (lkrnac.net).
Secondary site has its own child theme. In that child theme I used absolute paths for font:
@font-face {
font-family: 'Genericons';
src: url('https://mia.handysvandy.net/wp-stuff/themes/spacious-mia.handysvandy.net/font/genericons-regular-webfont.eot');
src: url('https://mia.handysvandy.net/wp-stuff/themes/spacious-mia.handysvandy.net/font/genericons-regular-webfont.eot?#iefix') format('embedded-opentype'),
url('https://mia.handysvandy.net/wp-stuff/themes/spacious-mia.handysvandy.net/font/genericons-regular-webfont.woff') format('woff'),
url('https://mia.handysvandy.net/wp-stuff/themes/spacious-mia.handysvandy.net/font/genericons-regular-webfont.ttf') format('truetype'),
url('https://mia.handysvandy.net/wp-stuff/themes/spacious-mia.handysvandy.net/font/genericons-regular-webfont.svg#genericonsregular') format('svg');
font-weight: normal;
font-style: normal;
}
Works fine now.